Ay's unimodality conjecture for rectangular Kronecker coefficients

Let $g(\lambda^{(1)},\ldots,\lambda^{(d)})$ denote the generalized Kronecker coefficient, where the arguments are partitions of the same integer, and write $g_d(n,k)=g(n\times k,\ldots,n\times k)$, with $n\times k=(k,\ldots,k)$ consisting of $n$ parts. A sequence is unimodal if it weakly increases and then weakly decreases. Ay's unimodality conjecture. Let $d\geq 3$ be odd and let $k$ be even. Then the sequence

$$\{g_d(n,k)\}_{n=0,\ldots,k^{d-1}}$$

is unimodal.

The sequence is symmetric for fixed $k$ and odd $d\geq 3$, and $g_d(n,k)=0$ for $n>k^{d-1}$. The conjecture is proved in this paper for $k=2$, while the general even-$k$ case remains open.