For trying to give a conclusive answer it seems necessary to rigorously characterize the geometry (the "conicidence structure", incl. the "light-cone structure") of the region under consideration (arguably with exception of "the singularity itself"). Unfortunately, this seems complicated (as may be gathered from efforts to tackle related problems at least approximately). Therefore the following only gives the outlines of an argument for a special case.

Let's consider one smartphone ($\mathsf A$) "falling freely" and "radially (towards the singularity)", and another smartphone ($\mathsf B$; "in the other hand", separated from $\mathsf A$) "moving radially" as well, and such that $\mathsf A$ and $\mathsf B$ remain "parallel (in the sense of Marzke-Wheeler)" throughout. (This condition can presumably be satisfied based on the otherwise undefined notions "radial" and "free fall", where the latter appears explicitly in the M-W definition of "parallelism", too.)

Further, Person ($\mathsf P$) shall move along such that throughout

• $\mathsf P$ finds coincident pings wrt. $\mathsf A$ and $\mathsf B$;
  in other words: for each of $\mathsf P$'s indications (such as any particular "facial expession" of $\mathsf P$) $\mathsf P$ observed/reviewed that smartphone $\mathsf A$ had observed and in turn displayed/reflected this indication of $\mathsf P$ and in coincidence $\mathsf P$ observed/reviewed that smartphone $\mathsf A$ had observed and in turn displayed/reflected this same indication of $\mathsf P$,

• $\mathsf A$ finds coincident pings wrt. $\mathsf P$ and $\mathsf B$;
  in other words: for each of $\mathsf A$'s indications (such as any particular "flash signal" of $\mathsf A$) $\mathsf A$ observed/took the picture (in coincidence) of both $\mathsf P$ and $\mathsf B$ reflecting this indication of $\mathsf A$, and likewise

• $\mathsf B$ finds coincident pings wrt. $\mathsf P$ and $\mathsf A$.

Moreover, let's require throughout (provided it can be satisfied at all) that $\mathsf P$ and $\mathsf A$ are M-W-parallel to each other, and that $\mathsf P$ and $\mathsf B$ are M-W-parallel to each other, too.

The Marzke-Wheeler construction of (the definition how to measure) "parallelism" of a pair of suitable participants involves reference to a certain set of events, such as the "reflection event off particle (II)" and the "reflection event off particle (III)" in this sketch of (the definition how to measure). If three participánts are pairwise M-W-parallel wrt. the same set of (at least several) events then let's call them "aligned to each other".

The point is: participants $\mathsf A$, $\mathsf B$ and $\mathsf P$, as described above (finding mutually coincident pings, and being pairwise M-W-parallel to each other) are not "aligned to each other". In other words, the configuration specified so far has $\mathsf A$ and $\mathsf B$ "falling one behind the other on the same radial track", while $\mathsf P$ is "moving along on the side", and possibly "rotating around $\mathsf A$'s and $\mathsf B$'s track".

Now, there may be certain additional participants identified in reference to $\mathsf A$, $\mathsf B$ and $\mathsf P$; namely:

• participant $\mathsf N$ such that any one among $\mathsf A$, $\mathsf B$, $\mathsf P$ and $\mathsf N$ finds coincident pings with respect to the three others; and likewise

• participant $\mathsf Q$, distinct and separate from $\mathsf N$, such that any one among $\mathsf A$, $\mathsf B$, $\mathsf P$ and $\mathsf Q$ finds coincident pings with respect to the three others.

Toegether, the specified configuration of the five participants $\mathsf A$, $\mathsf B$, $\mathsf N$, $\mathsf P$ and $\mathsf Q$ resembles that of five vertices of a (regular) triangular bi-pyramid (a.k.a. "(regular) triangular di-pyramid"), with $\mathsf N$ and $\mathsf Q$ corresponding to the two opposite "pyramid tips", and $\mathsf A$, $\mathsf B$ and $\mathsf P$ "at the waist".

In an actual regular triangular bi-pyramid (flat, non-rotating, in a flat region) the distance between its two opposite "pyramid tips" is of course equal to the $\sqrt{6}$-fold of the distance between any other pair of vertices.
Correspondingly it may be checked, for instance, whether

(1) $\mathsf N$ observed the completion of 2 consecutive "signal roundtrips" to and from $\mathsf Q$ before the completion of the corresponding 5 consecutive "signal roundtrips" to and from $\mathsf P$ (since $2~\sqrt{6} \lt 5$),

(2) $\mathsf N$ observed the completion of 20 consecutive "signal roundtrips" to and from $\mathsf Q$ before the completion of the corresponding 49 consecutive "signal roundtrips" to and from $\mathsf P$ (since $20~\sqrt{6} \lt 49$),

(3) $\mathsf N$ observed the completion of 9 consecutive "signal roundtrips" to and from $\mathsf Q$ after the completion of the corresponding 22 consecutive "signal roundtrips" to and from $\mathsf P$ (since $9~\sqrt{6} \gt 22$), etc.

Further, for each pair of the participants $\mathsf A$, $\mathsf B$, $\mathsf N$, $\mathsf P$ and $\mathsf Q$ it may be required (or at least be checked) whether an additional participant can be identified as "middle between" the pair under consideration; i.e. by finding coincident pings and by "alignment" as described above. For instance, participant "$\mathsf M[~\mathsf A, \mathsf B~]$" would be identified as the (unique) "middle between" $\mathsf A$ and $\mathsf B$ (throughout the entire trial) by

• $\mathsf M[~\mathsf A, \mathsf B~]$ finding for each indication coincident pings with respect to $\mathsf A$ and $\mathsf B$, and

• $\mathsf M[~\mathsf A, \mathsf B~]$, $\mathsf A$ and $\mathsf B$ being throughout aligned with respect to each other.

Comparing again to geometric relations in actual regular triangular bi-pyramid (flat, non-rotating, in a flat region) it may be checked moreover whether

(4) $\mathsf M[~\mathsf A, \mathsf P~]$ found coincident pings necessarily with respect to $\mathsf A$, $\mathsf P$,
but also with respect to $\mathsf M[~\mathsf A, \mathsf B~]$, $\mathsf M[~\mathsf B, \mathsf P~]$, $\mathsf M[~\mathsf N, \mathsf P~]$, $\mathsf M[~\mathsf P, \mathsf Q~]$, $\mathsf M[~\mathsf A, \mathsf N~]$, and $\mathsf M[~\mathsf A, \mathsf P~]$,

(5) $\mathsf M[~\mathsf N, \mathsf Q~]$ found coincident pings with respect to $\mathsf A$, $\mathsf B$, and $\mathsf P$,

(6) $\mathsf M[~\mathsf N, \mathsf Q~]$ found coincident pings with respect to $\mathsf M[~\mathsf A, \mathsf B~]$, $\mathsf M[~\mathsf A, \mathsf P~]$, and $\mathsf M[~\mathsf B, \mathsf P~]$, and

(7) $\mathsf M[~\mathsf N, \mathsf Q~]$ found the completion of any 1 "signal roundtrip" to and from $\mathsf P$ coincident to the completion of the corresponding 2 "signal roundtrips" to and from $\mathsf M[~\mathsf A, \mathsf B~]$.

Recalling that the light-cone structure in the region under consideration is complicated, it may be argued that

• the criteria (4 ... 7) may not (all) be found satisfied exactly; and satisfied at least approximately only in the limit as $\mathsf A$, $\mathsf B$ and $\mathsf P$ are not separated from each other, and

• by quantifying the possible deviations from the criteria (1 ... 7) being satisfied, or by similar/related measurements, the region containing $\mathsf A$, $\mathsf B$ and $\mathsf P$ may be characterized, as they are "falling". An applicable quantity of particular interest for this purpose is apparently (the sign of) "Karlhede's invariant", cmp. http://arxiv.org/abs/1404.1845 .