The generalized restricted-partition recurrence for unimodal polynomials

Let $G_s(n,k)$ be the recurrence obtained by summing over partitions of $k$ having size at most $s$, with $q$ an indeterminate and $n,k,s$ in the domain where this recurrence is defined. Generalized recurrence conjecture.

$$q^n(q^{k+s}-1)G_s(n+1,k)-q^{k+1}(q^n-q^{s-2})G_s(n,k+1)+(q^n-q^{k+s-1})G_s(n+1,k+1)=0.$$

The author reports having found this relation experimentally for $s=1,2,3,4$ and conjectures it for general $s$; no proof or resolution is given in the supplied text.