The switch-map symmetry for Hall scalar products

Let $m,n>0$, let $\mu=\mu_1\cdots\mu_k\vdash n$ be a composition as used in the paper, and define

$$\mu^{\vee}=(m-\mu_k)\cdots(m-\mu_1).$$

Let $\mathrm{Q}_{m,n}$ be the rational shuffle operator, $1$ the unit symmetric function, $h_\mu$ the complete homogeneous symmetric function indexed by $\mu$, and $\langle\,\ \rangle$ the Hall scalar product. The switch-map scalar-product conjecture.

$$\langle \mathrm{Q}_{m,n}(1),h_\mu\rangle=\langle \mathrm{Q}_{m,mk-n}(1),h_{\mu^{\vee}}\rangle.$$

The paper derives the coprime case from the switch-map properties; the full statement for arbitrary positive $m,n$ is presented as a conjecture.