Rationality conjectures for two harmonic WZ-seed dimension series

Let $S_1=S_1(a,b,c,d;n)$ and $S_2=S_2(a,b,c,d;n)$ be the two summands obtained from the WZ seeds in the harmonic extension of the displayed series, and let $V_N(S_i)$ be their degree-$N$ coefficient spaces. Two-seed rationality conjecture. The generating functions of the individual dimension sequences, and of the dimension sequence of their sum, are rational; more precisely,

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

and

$$\sum_{N\geq0}\dim V_N(S_2)t^N\stackrel{?}{=}\frac{1}{(1-t)^2(1-t^2)^2}.$$

The source also asserts rationality for the generating function of $\dim(V_N(S_1)+V_N(S_2))$, but does not provide its explicit form. These conjectures encode the observed number of independent harmonic-number summation identities; no proof is given.