Tilde-operation conjecture for products of skew Schur functions

Let $\mu/\alpha$ and $\nu/\beta$ be skew shapes. Define the tilde operation on pairs of outer and inner partitions, and write

$$(\mu,\nu)^\sim=(\lambda,\rho),\qquad (\alpha,\beta)^\sim=(\sigma,\tau).$$

Then set

$$(\mu/\alpha,\nu/\beta)^\sim=(\lambda/\sigma,\rho/\tau).$$

Tilde-operation conjecture for skew Schur functions. For all $\mu/\alpha$ and $\nu/\beta$, if $(\mu/\alpha,\nu/\beta)^\sim=(\lambda/\sigma,\rho/\tau)$, then

$$s_{\lambda/\sigma}s_{\rho/\tau}-s_{\mu/\alpha}s_{\nu/\beta}$$

is Schur-positive.

This is a proposed skew-shape generalization of the ordinary tilde-operation conjecture. The paper reports computer evidence and results in support of it, but does not establish it in full generality.