Rank-one orbifold instanton conjecture on $[\mathbb{C}^3/(\mathbb{Z}_2\times\mathbb{Z}_2)]\times\mathbb{C}$

Let $\vec{\mathtt{q}}=(\mathtt{q}_0,\mathtt{q}_1,\mathtt{q}_2,\mathtt{q}_3)$ and $\mathtt{Q}=\mathtt{q}_0\mathtt{q}_1\mathtt{q}_2\mathtt{q}_3$. Define

$$L(x_1,x_2,x_3,q)=\widetilde M(x_1,q)\widetilde M(x_2,q)\widetilde M(x_3,q)\widetilde M(x_1x_2x_3,q),$$

where $M(x,q)=\prod_{k=1}^{\infty}(1-xq^k)^{-k}$ and $\widetilde M(x,q)=M(x,q)M(x^{-1},q)$. For $p<s$ in $\{1,2,3\}$, let $(ps)^-$ be the unique element of $\{1,2,3\}\setminus\{p,s\}$. Rank-one $\mathbb{Z}_2\times\mathbb{Z}_2$ orbifold conjecture. The equivariant instanton partition function of the cohomological $\mathsf{U}(1)$ gauge theory with massive fundamental matter on $[\mathbb{C}^3/(\mathbb{Z}_2\times\mathbb{Z}_2)]\times\mathbb{C}$ is

\begin{aligned} Z&=\frac{M(\mathtt{Q})^{m\frac{\epsilon_1\epsilon_2\epsilon_3-\epsilon_1^2\epsilon_2-\epsilon_1^2\epsilon_3-\epsilon_2^2\epsilon_3-\epsilon_1\epsilon_2^2-\epsilon_1\epsilon_3^2-\epsilon_2\epsilon_3^2}{\epsilon_1\epsilon_2\epsilon_3\epsilon_4}}{L(-\mathtt{q}_1,-\mathtt{q}_2,-\mathtt{q}_3,\mathtt{Q})^{m/\epsilon_4}}\\ &\quad\times\prod_{1\leq p<s\leq3}\widetilde M(\mathtt{q}_p\mathtt{q}_s,\mathtt{Q})^{m\frac{\epsilon_{(ps)^-}-\epsilon_{ps}}{2\epsilon_4\epsilon_{(ps)^-}}}. \end{aligned}

The formula is motivated by dimensional reduction and matching leading terms with the known three-orbifold Donaldson--Thomas expression; a direct proof is not supplied, although the source notes that related techniques may apply.