Rationality conjectures for three harmonic extensions of a Ramanujan series

Let $S_1,S_2,S_3$ be the three summands obtained from the WZ seeds used to construct harmonic extensions of the displayed Ramanujan $1/\pi$-formula, and let $V_N(S_i)$ denote the span of their degree-$N$ parameter coefficients. Ramanujan-extension dimension conjecture. Each row of the dimension table has a rational generating function; specifically,

$$\sum_{N=0}^{\infty}\dim V_N(S_1)t^N\stackrel{?}{=}\frac{1-t+2t^2-t^3}{(1-t)^3(1-t^2)},$$ $$\sum_{N=0}^{\infty}\dim V_N(S_2)t^N\stackrel{?}{=}\frac{1}{(1-t)(1-t^2)^2(1-t^3)},$$

and

$$\sum_{N=0}^{\infty}\dim V_N(S_3)t^N\stackrel{?}{=}\frac{1}{(1-t)^2(1-t^2)^2}.$$

The conjecture summarizes observed dimensions of the independent summation identities generated by the three WZ seeds, including their combinations. It is based on finite computational data and is unresolved in the source.