Rationality conjectures for the coefficient and image dimensions of S_1

Let $S_1=S_1(a,b,c,d;n)$ be the first summand in the preceding pair of WZ constructions, let $V_N(S_1)$ be the span of the coefficients of degree-$N$ parameter monomials, and let

$$\Sigma:V_N(S_1)\longrightarrow\operatorname{MZV}_{N+2},\qquad f(n)\longmapsto\sum_{n\geq1}f(n).$$

S_1 dimension conjecture. The following two assertions are conjectured:

(a)

$$\sum_{N\geq0}\dim V_N(S_1)t^N\stackrel{?}{=}\frac{1}{(1-t)^3(1-t^2)^2}-\frac{t^2}{1-t}.$$

(b) The generating function

$$\sum_{N\geq0}\dim\Sigma(V_N(S_1))t^N$$

is a rational function of $t$.

The coefficient spaces generate rapidly increasing families of harmonic-number identities, while their summed images lie in multiple-zeta-value spaces whose conjectural dimensions grow exponentially. The source reports agreement with the first several computed dimensions but gives no proof of either assertion.