Rationality conjecture for the image series of the coefficient-summation map

Let $S=S(a,b,c,d,e;n)$ be the summand in the preceding WZ identity, let $V_N(S)$ be its degree-$N$ coefficient space, and define the linear summation map

$$\Sigma:V_N(S)\longrightarrow\operatorname{MZV}_{N+3},\qquad f(n)\longmapsto\sum_{n\geq1}f(n).$$

Image Hilbert–Poincaré conjecture. The generating function of the dimensions of the images is rational, with initial expansion

$$\sum_{N\geq0}\dim\Sigma(V_N(S))t^N\stackrel{?}{=}1+t+t^2+2t^3+2t^4+\cdots.$$

The map sends coefficient-extraction identities to multiple-zeta values; the displayed initial dimensions indicate that relations among the resulting sums are substantially stronger than relations among the coefficient spaces themselves. No rational form or proof is supplied in the source.