After being open for decades, Francisco Santos has recently proved that this fails in general.
Is it possible that the conjecture holds for $n < 2d$? Santos's counterexample had $(n,d) = (86, 43)$.
One observation which may be relevant: If $n < 2d$, then every pair of vertices has a common facet. One can use this to show that the general Hirsch conjecture reduces to the $n \ge 2d$ case, (see Ziegler's book Lectures on Polytopes, p. 84). But this doesn't seem to answer the question here.