The skew Hall–Littlewood rule for multiplication by an elementary symmetric function

Let $\lambda$ and $\mu$ be partitions with $\mu \subseteq \lambda$, and let $r\geq 0$. Write $e_r=P_{1^r}$. For a vertical strip $\alpha/\beta$, define

$$\operatorname{vs}(\alpha/\beta)=\prod_{i\geq1}\begin{bmatrix}\alpha_i^c-\alpha_{i+1}^c\alpha_i^c-\beta_i^c\end{bmatrix}_q,$$

where $\begin{bmatrix}n\k\end{bmatrix}_q$ is the $q$-binomial coefficient. The skew Hall–Littlewood–elementary-function conjecture.

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

where the sum is over all $\lambda^+\supseteq\lambda$ and $\mu^-\subseteq\mu$ such that $\lambda^+/\lambda$ and $(\mu/\mu^-)^c$ are vertical strips and $|\lambda^+/\lambda|+|\mu/\mu^-|=r$.

This is a conjectured skew version of the Hall–Littlewood Pieri rule. For $\mu=\emptyset$, it specializes to a standard Hall–Littlewood identity; the source gives no resolution of the general skew statement.