Kirillov–Klyachko saturation conjecture for reduced Kronecker coefficients

For partitions $\lambda,\mu,\nu$, let $\bar g_{\lambda\mu}^{\nu}$ denote the reduced (or stable) Kronecker coefficient. The saturation property concerns simultaneous scaling of the three partitions.

Kirillov–Klyachko conjecture. The reduced Kronecker coefficients satisfy

$$\bar g_{k\lambda,k\mu}^{k\nu}\ne0\text{ for some }k\geq1\quad\Longrightarrow\quad\bar g_{\lambda\mu}^{\nu}\ne0.$$

This conjecture was refuted in general: for every $k\geq3$, $(1^{k^2-1},1^{k^2-1},k^{k-1})$ is a counterexample. Thus the proposed saturation property fails, although special cases may remain valid.