Gessel's conjectured gamma-coefficient inequality

Let $D4AE_n$ denote the symmetric group on $n$ letters. Let $D4AE_n(231)$ be the set of permutations avoiding the pattern $231$, and let $S_n(t)$ be its descent polynomial. Write

$$S_n(t)=\sum_k \gamma^S_{n,k}t^k(1+t)^{n-1-2k}.$$

Let $D4AE_{n,i,j}$ be the coefficients in the expansion defining Gessel's gamma quantities. Gessel's conjectured inequality. For every $n$ and every $k$,

$$\gamma_{n,k,n-1-2k}\geq \gamma^S_{n,k}.$$

This inequality compares the gamma coefficients associated with Gessel's refinement to those of the descent polynomial on $231$-avoiding permutations. The source presents it as a conjectured inequality; its status is not established in the supplied text.