Minimal strong generating type conjecture for the N=2 coset ${\mathcal C}^{\psi}_{{\mathcal N}=2}(n,r|s)$

Let $n,r,s$ be non-negative integers, and define

$$\zeta(n,r|s):=\begin{cases}(1+s)(2+n+r+nr+rs),&r\geq s,\\(1+r)(1+n+2s+ns+rs),&r<s.\end{cases}$$

Let ${\mathcal C}^{\psi}_{{\mathcal N}=2}(n,r|s)$ be the corresponding coset vertex algebra. Minimal strong generating type conjecture. The algebra ${\mathcal C}^{\psi}_{{\mathcal N}=2}(n,r|s)$ has minimal strong generating type

$${\mathcal W}\bigg(1,2^2,3^2,\dots,(\zeta(n,r|s)-1)^2,\zeta(n,r|s);\bigg(\frac32\bigg)^2,\bigg(\frac52\bigg)^2,\dots,\bigg(\zeta(n,r|s)-\frac12\bigg)^2\bigg).$$

The claim is motivated by expected decoupling relations arising from invariant-theoretic relations, but the supplied context says it is not proved.