The f-vector comparison conjecture for order and chain polytopes

Let $P$ be a finite poset with $|P|=d>1$. The order polytope ${\mathcal O}(P)$ and chain polytope ${\mathcal C}(P)$ are the polytopes associated with $P$, and write their $f$-vectors as

$$f({\mathcal O}(P))=(f_0,f_1,\ldots,f_{d-1}),\qquad f({\mathcal C}(P))=(f'_0,f'_1,\ldots,f'_{d-1}).$$

The f-vector comparison conjecture. One has $f_i\leq f'_i$ for all $1\leq i\leq d-1$. Moreover, if $f_i=f'_i$ for some $1\leq i\leq d-1$, then ${\mathcal O}(P)$ and ${\mathcal C}(P)$ are unimodularly equivalent.

The claim compares the face numbers of the two naturally associated polytopes. The supplied text does not give evidence that this statement has been resolved, so its status remains open.