Rationality conjecture for the coefficient dimension series of a zeta-value WZ-seed extension

Let $S=S(a,b,c,d,e;n)$ be the summand obtained from the WZ-seed extension of the displayed $\zeta(5)$ series, and let $V_N(S)$ be the span of the coefficients of degree-$N$ parameter monomials. Zeta-extension dimension conjecture. The generating function of these dimensions is conjectured to be

$$\sum_{N\geq0}\dim V_N(S)t^N\stackrel{?}{=}\frac{1}{(1-t)(1-t^2)(1-t^3)(1-t^4)(1-t^5)}.$$

The initial coefficients are $1+t+2t^2+3t^3+5t^4+7t^5+10t^6+\cdots$. The conjecture predicts a polynomially generated pattern for the independent harmonic-number identities arising from this five-parameter extension; the source supplies no proof.