The conjecture for $\mathbf{Q}_{m,n}^{k}(1)$

Let $M$ and $N$ be positive integers, and define

$$k=\gcd(M,N),\qquad m=M/k,\qquad n=N/k.$$

Let $L_{M,N}$ be the symmetric function associated with the $M,N$-path tuples, let $\mathbf{Q}_{m,n}$ be the elliptic Hall algebra operator, and let $\operatorname{pdinv}(\bm{P})$ denote the path statistic used in the definition of $L_{M,N}$. Define $C$ by

$$C=\max_{\bm{P}}\operatorname{pdinv}(\bm{P}),$$

where the maximum is over all $M,N$-path tuples $\bm{P}$. The conjecture for $\mathbf{Q}_{m,n}^{k}(1)$. One has

$$\mathbf{Q}_{m,n}^{k}(1)=\pm(1-q)^k t^C L_{M,N}.$$

This is the paper's main conjecture, relating the elliptic Hall algebra expression to the lattice-path symmetric function; the supplied material does not state whether it has been resolved.