The concise symmetric Kronecker coefficient conjecture

For a partition $\lambda$, let $g(\lambda,\mu,\nu)$ be the Kronecker coefficient and define the symmetric Kronecker coefficient by

$$g_s(\lambda)=g(\lambda,\lambda,\lambda).$$

A counting function is concise if every positive integer $k$ occurs as a value on an input whose size is bounded by a fixed polynomial in $\log k$. Symmetric Kronecker coefficient conjecture. The function $g_s$ is concise. The function is known to be complete and its maximum on partitions of size $n$ has exponential order, but the asserted polynomial-size realization of every value remains open.