Littlewood–Richardson coefficient monotonicity under the starred transformation

Given an ordered pair $(\lambda,\mu)$ of partitions with the same number of parts, define

$$\lambda^*_k=\lambda_k-k+\#\{\ell\mid \mu_\ell-\ell\geq\lambda_k-k\},$$

and

$$\mu^*_\ell=\mu_\ell-\ell+1+\#\{k\mid \lambda_k-k>\mu_\ell-\ell\}.$$

For any partition $\nu$, the starred-transformation conjecture.

$$c_{\,\lambda\,\mu}^{\,\nu}\leq c_{\,\lambda^*\,\mu^*}^{\,\nu}.$$

This is a combinatorial conjecture about Littlewood–Richardson coefficients; the supplied text gives no resolution, and the surrounding discussion presents it as an open problem arising from a geometric approach.