Rationality conjecture for the Hilbert–Poincaré series of coefficients of a very-well-poised hypergeometric series

Let

$$f(a,b,c,d,e):=\sum_{k\geq 0}\frac{(a+2e+2k+2)(a+e+1)_k(b+e+1)_k(c+e+1)_k(d+e+1)_k}{(e+1)_{k+1}(a-b+e+1)_{k+1}(a-c+e+1)_{k+1}(a-d+e+1)_{k+1}}.$$

Write its power-series expansion near $(0,0,0,0,0)$ as

$$f(a,b,c,d,e)=\sum_{i_1,\ldots,i_5\geq0}c(i_1,\ldots,i_5)a^{i_1}b^{i_2}c^{i_3}d^{i_4}e^{i_5},$$

and define

$$V_n:=\operatorname{Span}_{\mathbb Q}\{c(i_1,\ldots,i_5)\mid i_1+\cdots+i_5=n\}.$$

Hilbert–Poincaré rationality conjecture. The Hilbert–Poincaré series of these coefficient spaces is rational:

$$\sum_{n=0}^{\infty}\dim_{\mathbb Q}V_n\,t^n\in\mathbb Q(t).$$

The coefficients have weight $2+n$, so these spaces organize the multiple-zeta values produced by coefficient extraction from the hypergeometric series. The conjecture concerns an underlying graded-algebra structure suggested by the observed simple form of the generating series; no resolution is supplied in the source.