Power-sum Laurent integrality for the Macdonald shift map

Let $p_\mu$ be a power-sum symmetric function, let $P^\circ_\nu$ be the shifted power-sum basis, and define

$$b_{\mu\nu}:=[P^\circ_\nu]\operatorname{sh}_{q,t}(p_\mu)\in\mathbb{Q}(q,t),$$

where $\operatorname{sh}_{q,t}$ is Lassalle's Macdonald shift map. Power-sum Laurent-integrality conjecture. The coefficient $b_{\mu\nu}$ is Laurent:

$$b_{\mu\nu}\in\mathbb{Q}[q^{\pm1},t^{\pm1}],$$

and in fact

$$b_{\mu\nu}\in\mathbb{Q}[q,t^{\pm1}].$$

This is the power-sum-basis form of the conjectured Laurent-lattice isomorphism for the shift map. It remains open; the source contrasts it with Knop's corresponding theorem for integral Macdonald functions.