Universal-function numerical conjecture for virtual refined invariants

Let $\underline{\beta}=(\beta_1,\beta_2,\beta_3,\beta_4)\in\mathbb Z^4$ satisfy $\beta_1\equiv\beta_2\pmod 2$ and $\beta_3\geq\beta_4-3$, and let $n<\frac12(\beta_1-\beta_2)+2\beta_4$. Let $(\gamma_1,\gamma_2)\in\mathbb Z^2$, and let $\mathsf A_{\underline\alpha}$ denote the universal function defined from the seven universal series in the paper. Define

$$\phi(x,y):=\prod_{m=1}^{\infty}\frac{1}{(1-x^{2m})^{10}(1-x^{2m}y)(1-x^{2m}y^{-1})}.$$

Universal-function numerical conjecture. The coefficient of $x^{4n-\beta_1-3\beta_4}$ in the $s^0$-coefficient of the indicated sum of the two universal-function terms equals the coefficient of $x^{4n-\beta_1-3\beta_4}$ in

$$8(-1)^{\gamma_2}\left(\frac{\phi(x,y)}2\right)^{\beta_4}\left(\frac{2\overline\eta(x^4)^2}{\theta_3(x,y^{1/2})}\right)^{\beta_3}\left(\frac{\theta_3(x,y^{1/2})}{\theta_3(-x,y^{1/2})}\right)^{\gamma_1},$$

where the two universal-function terms are exactly those displayed in the source. This conjecture predicts the numerical identities required for the proposed universal formula and is supported by computer calculations, but remains open.