Planck distance in different aeons in Penrose's Cyclic Conformal Cosmology From a layman perspective, I understand that conformal geometry can explain why the immense size of the universe in one aeon at its end can be equivalent to the incredibly small size of a new universe that is about to have a Big Bang in the next aeon. In conformal geometry scale does not matter.
From one aeon to the other the fundamental constants of nature could have different values, but is it true that Planck's distance is one constant that have necessarily to change value, because the entire dimension of a preceding universe could fit inside the Planck distance of the next universe?
 A: Whether different regions of spacetime should be considered to be separate "universes" is a question that Penrose's much-disliked (but possibly accurate) Conformal Cyclic Cosmology may push to the current limit.
The explicitly multiverse models that I've seen in the scientific literature are all "inflationary" ones requiring an asymptotically-infinite spatial expansion, and Penrose has very firmly opposed all notions of such cosmic inflation as being in conflict with entropy, which he feels to have a gravitational version that works in a sense inverse to the thermodynamic one upon which our own impression of passage through time is based. (That time does have a thermodynamic basis is a view that Penrose shares with nearly all other physicists:  The only exception I've noticed has been another mathematical physicist, Julian Barbour, who feels that time has a gravitational basis.)
"Aeons" are extremely and indefinitely prolonged periods of time, but the separations between Penrose's aeons are, in fact, characterized by an absence of measurable or observable passage through it, as all of the matter which might have occupied them has evaporated into radiation, leaving those separations without the rest mass that's essential for the construction, formation, or duration of any real or imaginable "clocks", either natural or artificial:  The last of that radiation to form is considered, by him and other physicists, to be the "Hawking radiation" emitted through quantum processes near the event horizons of black holes.  (However, although many quantum physicists feel that the "unitarity" implied by Schrodinger's equation would allow some unspecified analysis of Hawking radiation to reconstruct the reality of all the material ingested by the black holes formed over the course of an aeon, Penrose does not share that view, and considers that a singular volume within every black hole--not necessarily a point, as a singularity's usually described, but possibly a spatial volume--effectively nullifies the ingested material, leaving it without any remaining role in the dynamics of the universe.)
The reality Penrose's model describes may lack that variety which might characterize the cosmic inflation that's based on a hypothetical scalar field, but the spatial connection between aeons gives Penrose's model an unusually wide applicability, which has resulted in some observational support for it:  Radiation from the black holes of previous aeons may remain the best explanation for spots of "significantly raised temperature" in the Cosmic Microwave Background radiation, as described at https://arxiv.org/abs/1808.01740  .
The idea that a Planck length within the aeon immediately previous to our own might have provided the space for the full extent of our own aeon makes an unjustified assumption:  In fact, the possibility that a Planck length within our own aeon might have accomodated the full spatial extent of the previous aeon seems equally consistent with the Weyl Curvature Hypothesis (preserving angles but not lengths) that provides the basis for Penrose's model.
The OP's preference for the first-mentioned of those two possibilities (which, to use physics' terminology, is the "well-defined" one) provides a sensible reflection of the fact that, although we can see neither the future nor necessarily all of the past, we can see the most deeply into the past through starlight and its red or blue shifts (providing observational evidence either for universal expansion or for an overall acceleration or deceleration of the velocities at which the motion of stars is generally widening or--at least hypothetically--narrowing the space between them), as well as through the CMB radiation.
As I've tried to make clear here, the Weyl Curvature Hypothesis would be valid either for universal expansion or for universal contraction, in which we would participate, if only to a cosmically infinitesimal extent.
However, it might be noted that, as described by Rutgers University's Lindford in his paper "Big Bounce or Double Bang?" (whose preprint is freely visible on Arxiv), the entropic arrow of time (which is consistent with the feelings of biological beings during their passage through it) is felt to reverse, due to the "clumping" effects of gravity, at the interface between any pair of Penrose's temporal aeons, whereas the causal arrow of time is, at least in Lindford's assessment of it, felt to pass through all of the presumably infinite number of those aeons.  (The distinction between the two "arrows" may be subtle, but allows us to visualize events very far beyond our own lifespans or travels, and has vastly extended astronomical predictability, without making us "feel" 6,000,000,000 years old:  Whether the Planck length would maintain its naturalistic value throughout such a lengthy passage of time is, given the fact that Penrose feels that quantum fluctuations would tend to prevent any one aeon from being a carbon copy of any other, something I'd hate to bet on, however invaluable it may be for projecting the limits of energy available for telescopic magnification, OVNI or rocket travel, etc.)
