Rational Dyck path generating-series conjecture for torus links

Let $N$ and $M$ be positive integers, write $d=\frac{\gcd(N,M)}{}$, $N=dn$, and $M=dm$ with $n$ and $m$ coprime, and define

$$C_{N,M}(q,t)=\sum_{\Delta\in \mathbf{M}_{N,M}}q^{\mathtt{gap}(\Delta)}t^{\operatorname{dinv}(\Delta)}.$$

Let $P_{n,m}$ be the operator from the coprime case, and let $P_{n,m}^{d}$ denote its $d$-fold version. Generalized rational shuffle conjecture. For general $d\geq 1$,

$$C_{N,M}(q,t)=\frac{1}{(1-q)^{d-1}}(P_{n,m}^{d}(1),h_N),$$

and the series $C_{N,M}(q,t)/(1-q)$ agrees with the Poincaré series of the $(a=0)$ part of the Khovanov–Rozansky homology of the $(N,M)$ torus link. In the case $M=N$, part (a) is equivalent to a conjecture cited by the source and is stated there as still open; the claim extends the coprime rational shuffle and homological formulas to the non-relatively-prime case.