Observational status of Sciama's hypothesis I have always taken the existence of inertia more or less for granted, as an observational fact that does not require explanation.
But on reflection this is an unscientific attitude, and perhaps there exists a deeper reason for the existence of inertial mass. Of course, in the absence of an explanatory theory of inertia that makes testable predictions we should be wary of ascribing importance to an observation that seems to stand by itself, but that does not mean the question is somehow beyond the realm of scientific inquiry.
Happily, in (1) Sciama put forth the bold hypothesis that the inertia of a single object is due to the action of the mass of the rest of the universe (since becoming aware of this I have found various other theories of inertia of greater or lesser cogency, but many of them seem to veer into quackery).
Sciama also worked out a prediction of his theory - that is, his theory is falsifiable. Specifically, the gravitational constant becomes a function of the distribution of matter in the (presumably observable) universe, so that a precise value of G predicts a value for the mean density of the universe.
The value provided in the original paper of 1953 is $\rho \approx 5\times 10^{-27}g cm^{-3}  $ , which Sciama argued was not incompatible with the observational estimates of the time ($\rho \approx \times 10^{-30}g cm^{-3}$).
A quick google search (e.g. https://wmap.gsfc.nasa.gov/universe/uni_matter.html) suggests current estimates are around
$9.9 \times 10^{-30}gcm{^-3}$, i.e. still of the same order of magnitude as in the 1950s(!).
Is this sufficient to definitively falsify Sciama's theory (which made numerous simplifications), or are there reasons to doubt this quantity?
(1) Sciama, Dennis William. "On the origin of inertia." Monthly Notices of the Royal Astronomical Society 113.1 (1953): 34-42. https://doi.org/10.1093/mnras/113.1.34
 A: Summary:
Sciama's approach essentially ends up predicting the critical density of the Universe, so in that sense, it is entirely consistent with modern observation. Whether his theory actually brings any useful modern insight is quite another question.
TLDR version
This 2009 paper by Gine (you can find a pdf by searching Google Scholar) reproduces key elements of the Sciama paper in a more digestible fashion, pointing out that $\tau=1/H_0$ where $H_0$ is the Hubble Constant, with the key equation obtained by Sciama (Eqn. 4 in Gine's paper), with $R_u$ and $M_u$ being the radius and mass of the Universe:
\begin{equation}
\notag
G = \frac{c^2R_u}{M_u}
\end{equation}
Assuming a uniform Universal density $\rho_u$, and a sameness between inertial and gravitational mass, Gine presents Scima's Universal density estimate as:
\begin{equation}
\notag
\frac{4\pi G\rho_u}{3H_0^2}=1
\end{equation}
That should be recognizable as the critical density of the Universe $\rho_u \sim \rho_c$, although we seem to be missing a factor of two! Indeed we are, because while Gine kinda correctly associated $R_u$ with a universal radius, in fact $R_u=l_\Lambda$ i.e. the future cosmic event horizon radius, and $M_u=2M_{CEH}$, where $M_{CEH}$ is the rest mass-energy of the future cosmic event horizon, as per Lineweaver.
Therefore $2\rho_u=\rho_c=3H_0^2/8\pi G$. The critical density of the Universe.
