Graham's log-concavity conjecture for order-preserving maps

Let $P=(X,\prec)$ be a finite poset, let $x\in X$, and let $a,t\in\mathbb N_{\geq 1}$ with $1<a<t$. Write $\Omega(P,t;x,a)$ for the number of order-preserving maps $g:X\to[t]$ satisfying $g(x)=a$. Graham's conjecture. For these parameters,

$$\Omega(P,t;x,a)^2\geq\Omega(P,t;x,a+1)\cdot\Omega(P,t;x,a-1).$$

This is a pointwise log-concavity assertion for the distribution of the value at a fixed element of a poset. The source attributes it to Graham and does not give evidence of a resolution.