The generating-function conjecture for identities from a five-parameter hypergeometric extension

Let $S=S(a,b,c,d,e;n)$ denote the summand in the five-parameter hypergeometric extension described above, and let $V_N(S)$ be the space of independent summation formulas generated by $S$ from degree-$N$ monomials in its parameters. The coefficient of a degree-$N$ monomial on the right-hand side lies in the multiple zeta values of weight $N+3$.

Generating-function conjecture. The dimensions of these spaces satisfy

$$\sum_{N\geq 0} \dim V_N(S)t^N=\frac{1}{(1-t)(1-t^2)(1-t^3)(1-t^4)(1-t^5)}.$$

The displayed values for $N=0,\ldots,7$ are $1,1,2,3,5,7,10,13$. The conjecture predicts the number of independent summation formulas generated by this hypergeometric seed; its status is resolved according to the supplied evidence.