Rationality conjecture for the coefficient dimension series of a WZ-seed extension

Let $S=S(a,b,c;n)$ be the summand obtained from the WZ pair in the harmonic extension of the displayed Ramanujan $1/\pi$-series, and let $V_N(S)$ be the span of the coefficients of degree-$N$ parameter monomials. WZ-seed coefficient conjecture. The generating function of the coefficient-space dimensions is conjectured to be

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

Its initial values are $1+t+2t^2+2t^3+4t^4+4t^5+6t^6+\cdots$. This predicts a simple graded structure for the independent identities obtained by harmonic coefficient extraction from the WZ seed; the source gives only computational evidence.