Generalization of the Amdeberhan–Shareshian–Stanley conjecture for partition division maps

Let $\mu\vdash kn$. The symmetric function $\operatorname{rowDiv}_k(\mathrm{e}_\mu)$ is $\mathrm{e}$-positive with non-negative integer coefficients.

Generalized Amdeberhan–Shareshian–Stanley conjecture. For every partition $\mu$ of $kn$, the symmetric function obtained by applying the partition division map $\operatorname{rowDiv}_k$ to the elementary symmetric function $\mathrm{e}_\mu$ has a non-negative integer expansion in the elementary basis.

This conjecture extends the Amdeberhan–Shareshian–Stanley positivity phenomenon for partition division maps. The displayed computation provides an example, but the general assertion remains unresolved.