Positivity conjecture for rectangular DAHA-superpolynomials

Restrict to a type-$A$ root system and let $H\!D_{r,s}(\lambda;q,t,a)$ be the DAHA-superpolynomial. For a rectangular Young diagram of size $i\times j$, equivalently the dominant weight $j\omega_i\in P_+$, consider $H\!D_{r,s}(j\omega_i;q,t,a)$. The DAHA-to-DGR transformation is

$$t\mapsto q^{2},\qquad q\mapsto q^2t^2,\qquad a\mapsto a^2t.$$

Rectangular DAHA-superpolynomial positivity conjecture. The coefficients of $H\!D_{r,s}(j\omega_i;q,t,a)$ are positive integers. After the displayed transformation, the resulting polynomial recovers the superpolynomials from the cited literature.

This conjecture extends an earlier conjecture and connects DAHA-superpolynomials with the DGR convention for superpolynomials. The source does not state a resolution.