Ay's generalized unimodality conjecture for Kronecker coefficients

Let $g(\lambda^{(1)},\ldots,\lambda^{(d)})$ be the generalized Kronecker coefficient, let $\rho_k(\lambda)=(k,\lambda)$, and apply $\rho_k$ componentwise to tuples of partitions. Let $d\geq 3$ be odd, let $k$ be even, and let $\pmb{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(d)})$ be a $d$-tuple of partitions of $m\leq k^d/2$, each with parts at most $k$. Define

$$a=\max_i(\lambda^{(i)})'_1,\qquad b=\min_i(\lambda^{(i)})'_k.$$

Here $(\cdot)'$ denotes conjugation of a partition. Ay's generalized unimodality conjecture. The sequence

$$\{g(\rho_k^n\pmb{\lambda})\}_{n=-b,\ldots,k^{d-1}-a}$$

is unimodal.

This refines the rectangular case by allowing arbitrary tuples of partitions satisfying the stated bounds. The paper proves the corresponding claims for $k=2$, but the conjecture for general even $k$ remains open.