Rationality conjectures for three WZ-seed coefficient series

Let $S_1,S_2,S_3$ be the three summands defined from the WZ identities in the preceding construction, and let $V_N(S_i)$ denote the span of the coefficients of degree-$N$ parameter monomials in $S_i$. Three-seed dimension conjecture. The generating functions of the rows of the dimension table are rational; in particular,

$$\sum_{N\geq0}\dim V_N(S_1)t^N\stackrel{?}{=}\frac{1}{(1-t)^4(1-t^2)},$$

and

$$\sum_{N\geq0}\dim V_N(S_3)t^N\stackrel{?}{=}\frac{1}{(1-t)^3(1-t^2)}.$$

The claim concerns the number of independent summation formulas generated separately by the three related WZ seeds and by their combinations. It is an empirical rationality pattern based on the displayed initial dimensions; no proof is supplied in the source.