Fully symmetric Kronecker coefficient asymptotics

For a partition $\lambda\vdash n$, write $\lambda'$ for its conjugate partition, and define the maximal fully symmetric Kronecker coefficient by

$$K^{\mathrm{fs}}(n):=\max\{g(\lambda,\lambda,\lambda):\lambda\vdash n,\ \lambda=\lambda'\}.$$

Fully symmetric Kronecker asymptotic conjecture.

$$\log K^{\mathrm{fs}}(n)=\frac12 n\log n-O(n).$$

This conjecture predicts that fully symmetric Kronecker coefficients have the same leading logarithmic growth as the unrestricted maximum. The paper notes that only a nontrivial lower bound for the symmetric, rather than fully symmetric, maximum was previously known.