Refined Rogers–Ramanujan product conjecture

Let $q$ and $x$ be formal variables, and for each fixed integer $m\geq1$ let $n_{m,k}$ be positive integers indexed by finitely many $k\in I_m$. Refined Rogers–Ramanujan conjecture. For each fixed $m\geq1$, there exist finitely many such positive integers satisfying

$$\sum_{n\geq0}\frac{q^{n^2}}{(1-q)\cdots(1-q^n)}(q^{-1/2}x)^n=\prod_{m\geq1}\prod_{k\in\mathbb{Z}}\prod_{l\geq0}\left(1-q^{(k+1)/2+l}x^m\right)^{(-1)^m n_{m,k}}.$$

At $x=q^{1/2}$ and $x=q^{3/2}$, the identity yields the two Rogers–Ramanujan identities. It is proposed after computations in the framing-$\tau=1$ case and is connected to the unresolved positive-framing integrality problem.