Nekrasov's conjecture for the empty fourfold DT vertex

Let $t_1,t_2,t_3,t_4,y,q$ be formal variables satisfying $t_1t_2t_3t_4=1$. For any formal variable $x$, write $[x]=x^{\frac12}-x^{-\frac12}$, and let $\operatorname{Exp}$ denote the plethystic exponential. The empty fourfold Donaldson–Thomas vertex is $\mathsf{V}_{\varnothing\varnothing\varnothing\varnothing}^{\mathrm{DT}}(t,y,q)$. Nekrasov's conjecture. There exist unique choices of signs such that

$$\mathsf{V}_{\varnothing\varnothing\varnothing\varnothing}^{\mathrm{DT}}(t,y,q)=\operatorname{Exp}(\mathcal F(t,y,q)),$$

where

$$\mathcal F(t,y,q)=\frac{[t_1t_2][t_1t_3][t_2t_3][y]}{[t_1][t_2][t_3][t_4][y^{\frac12}q][y^{\frac12}q^{-1}]}\in \frac{\mathbb Q(t_1,t_2,t_3,t_4,y^{\frac12})}{(t_1t_2t_3t_4-1)}((q)).$$

The existence part is attributed to Nekrasov, while the uniqueness of the signs is the part conjectured here.