A genus-one FJRW invariant generating-function conjecture

Let $f_{d;k}^{(i)}$ be the coefficients arising from the localization and MSP-vanishing calculations, let $I_{i-1}(t)$ denote the corresponding genus-zero hypergeometric series, and let $i\in\{1,2,3,4\}$. The indices satisfy $d\geq 0$ and the displayed expressions involve the coefficient index $k$.

Generating-function conjecture. According to the established cases $i=1,2$, one conjectures that

$$\sum_{d=0}^{\infty}f_{d;i-3}^{(i)}t^{5d+i-1}=5^{2-i}I_{i-1}(t).$$

Moreover, the coefficient identity that would verify the conjecture is

$$\frac{f_{d,k}^{(i)}}{(5k+i)^{d+i-3}}=(-1)^{d-k}\binom{d}{k}\frac{f_{d,d}^{(2)}}{(5d+2)^{d+i-3}}.$$

The first two cases follow from the preceding formulas for $I_0(t)$ and $I_1(t)$; the cases $i=3,4$ remain to be verified by establishing the stated coefficient relation.