Rogers--Ramanujan identity for the infinite-rank series

Prove that, for coprime positive integers a,b, the infinite-rank series N_{a,b;\infty}(q,1) equals the explicit infinite product P_{a,b}(q)=\prod_{i\geq1}(1-q^i)^{-r_{a,b}(i)}, equivalently the normalized W_a(a,a+b)-character associated with the Euclidean rhythm.

Background

The polynomial and series N_{a,b;n}(q,t) encode motivic information about Quot schemes of torus-knot singularities. At infinite rank and after specializing t=1, the authors observe a product expansion whose exponents are periodic modulo a+b. They identify this product with a normalized character of a minimal-model W-algebra and emphasize that the identity appears to be deep and unproved in general.