Stanley's log-concavity conjecture for h-vectors of graded Cohen–Macaulay domains

Let $R=R_0\oplus R_1\oplus\dots$ be a graded (Noetherian) Cohen–Macaulay (or perhaps Gorenstein) domain over a field $K=R_0$, generated by $R_1$, with Krull dimension $d$. Let $H(R,m)=\dim_K R_m$ be its Hilbert function, and write

$$\sum_{m\ge0}H(R,m)x^m=(1-x)^{-d}\sum_{i=0}^s h_i x^i.$$

Stanley's conjecture. The sequence $(h_0,h_1,\dots,h_s)$ is log-concave, meaning that $h_{i-1}h_{i+1}\le h_i^2$ for $2\le i\le s-1$.

Stanley’s conjecture is false in general, as counterexamples are known; natural weakenings remain open. The paper proves a special case for ladder determinantal rings cogenerated by $2\times2$ minors.