Unimodality conjecture for the Kronecker-coefficient differences

For a natural number $n$, let $b_{\lambda,i}$ be the tableau multiplicities and let $g_{\lambda\mu}^{\nu}$ be the Kronecker coefficients. Define

$$d_{\nu,i}:=\sum_{\lambda\vdash n}\sum_{\mu\vdash n}b_{\lambda,i}b_{\mu,i}g_{\lambda\mu}^{\nu}-\sum_{\lambda\vdash n}\sum_{\mu\vdash n}b_{\lambda,i-1}b_{\mu,i+1}g_{\lambda\mu}^{\nu}.$$

Unimodality conjecture. For every $1\leq i\leq\binom{n}{2}-1$, the sequence $d_{\nu,i}$ is symmetric and unimodal. This is proposed as a stronger conjecture implying equivariant log-concavity; the available computations support it, but it remains unproved in general.