The skew Hall–Littlewood rule for multiplication by a Schur function

Let $\lambda$ and $\mu$ be partitions with $\mu \subseteq \lambda$, and let $r\geq 0$. For a skew shape $\alpha/\beta$, write $\operatorname{sk}(\alpha/\beta)$ for the coefficient defined in the source. A vertical strip is a skew shape with at most one cell in each row. The skew Hall–Littlewood–Schur conjecture.

$$P_{\lambda/\mu}\cdot s_r=\sum (-1)^{|\mu/\mu^-|}\operatorname{sk}(\lambda^+/\lambda)P_{\lambda^+/\mu^-},$$

where the sum is over all $\lambda^+\supseteq\lambda$ and $\mu^-\subseteq\mu$ such that $\mu/\mu^-$ is a vertical strip and $|\lambda^+/\lambda|+|\mu/\mu^-|=r$.

This is one of the conjectured skew analogues of the Hall–Littlewood Pieri rules; its specialization at $\lambda=\mu=\emptyset$ agrees with a known Hall–Littlewood identity. The source does not provide a resolution.