Infinitesimal change of stable basis

The purpose of this note is to study the Maulik-Okounkov $K-$theoretic stable basis for the Hilbert scheme of points on the plane, which depends on a "slope" $m \in \mathbb{R}$. When $m = \frac ab$ is rational, we study the change of stable matrix from slope $m-\varepsilon$ to $m+\varepsilon$ for small $\varepsilon>0$, and conjecture that it is related to the Leclerc-Thibon conjugation in the $q-$Fock space for $U_q\widehat{\mathfrak{gl}}_b$. This is part of a wide framework of connections involving derived categories of quantized Hilbert schemes, modules for rational Cherednik algebras and Hecke algebras at roots of unity.