Conjectural action of the spectral-dual generators on integral generalized Macdonald polynomials

Let $N$ be a positive integer, let $\vec{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(N)})$ be an $N$-tuple of Young diagrams, and let $\widetilde{M}_{\vec{\lambda}}$ be their integral generalized Macdonald polynomials. For a tuple $\vec{\lambda}$, write $A(\vec{\lambda})$ and $R(\vec{\lambda})$ for the addable and removable boxes, respectively, and for a box $x=(\ell,i,j)$ set $\chi_x=u_\ell t^{-i+1}q^{j-1}$. If $\vec{\mu}$ is obtained from $\vec{\lambda}$ by removing or adding one box, define

$$\tilde{c}^{(+)}_{\vec{\lambda},\vec{\mu}}=\xi^{(+)}_x\frac{\prod_{y\in A(\vec{\lambda})}(1-\chi_x\chi_y^{-1}(q/t))}{\prod_{\substack{y\in R(\vec{\lambda})\y\ne x}}(1-\chi_x\chi_y^{-1})},\qquad x\in\vec{\lambda}\setminus\vec{\mu},$$ $$\tilde{c}^{(-)}_{\vec{\lambda},\vec{\mu}}=\xi^{(-)}_x\frac{\prod_{y\in R(\vec{\lambda})}(1-\chi_y\chi_x^{-1}(q/t))}{\prod_{\substack{y\in A(\vec{\lambda})\y\ne x}}(1-\chi_y\chi_x^{-1})},\qquad x\in\vec{\mu}\setminus\vec{\lambda},$$

where

$$\xi^{(+)}_{(\ell,i,j)}=(-1)^{N+\ell}p^{-\frac{\ell+1}{2}}t^{(N-\ell)i}q^{(\ell-N+1)j}\frac{\prod_{k=1}^{N-\ell}u_{\ell+k}}{u_\ell^{N-\ell-1}},\qquad \xi^{(-)}_{(\ell,i,j)}=(-1)^\ell p^{\frac{\ell-1}{2}}t^{(\ell-2)i}q^{(1-\ell)j}\frac{\prod_{k=1}^{\ell-1}u_k}{u_\ell^{\ell-2}}.$$

Conjectural action of $X^{(1)}_{\pm1}$.

$$X^{(1)}_1\widetilde{M}_{\vec{\lambda}}\overset{?}{=}\sum_{\substack{|\vec{\mu}|=|\vec{\lambda}|-1\vec{\lambda}\supset\vec{\mu}}}\tilde{c}^{(+)}_{\vec{\lambda},\vec{\mu}}\widetilde{M}_{\vec{\mu}},\qquad X^{(1)}_{-1}\widetilde{M}_{\vec{\lambda}}\overset{?}{=}\sum_{\substack{|\vec{\mu}|=|\vec{\lambda}|+1\vec{\lambda}\subset\vec{\mu}}}\tilde{c}^{(-)}_{\vec{\lambda},\vec{\mu}}\widetilde{M}_{\vec{\mu}}.$$

This conjecture describes the spectral-dual action of the DIM algebra generators on the integral generalized Macdonald basis. The $X^{(1)}_1$ case and the displayed formulas have been checked computationally in small sizes, while the general action remains open.