A combined integrality conjecture for generalized Frobenius partition functions

Let $\alpha$ and $a$ be as in the preceding construction, let $\beta\in\{0,1\}$, let $\varphi$ denote the relevant theta function, let $q$ be the nome, let $W\tau$ denote the transformed parameter, and let $L_{\alpha}^{(\beta)}$ be the associated modular function. The preceding theorem shows that each summand, after multiplication by $\varphi\left((-1)^{\beta+1}q^{1+4a}\right)$, has coefficients supported in the appropriate residue class modulo $4$. Combined integrality conjecture.

$$\sum_{\beta=0,1}\varphi\left((-1)^{\beta+1}q^{1+4a}\right) L_{\alpha}^{(\beta)}(W\tau) \in\mathbb{Z}[[q^4]].$$

This conjecture proposes that summing the two $\beta$-components eliminates the remaining terms outside powers divisible by $4$. It is presented as a possible simpler statement following the preceding theorem, and no proof or resolution is supplied.