Nekrasov's sign-uniqueness conjecture for the zero-leg DT vertex

Let $\lambda_1,\lambda_2,\lambda_3$ be the equivariant parameters for the torus action on $\mathbb C^4$, with the Calabi–Yau relation determining the fourth parameter, and let $\mathsf V_{\varnothing\varnothing\varnothing\varnothing}^{\operatorname{DT}}(q)$ be the zero-leg DT vertex. Nekrasov's sign-uniqueness conjecture. There exist unique choices of signs such that

$$\mathsf V_{\varnothing\varnothing\varnothing\varnothing}^{\operatorname{DT}}(q)=\exp\left(q\frac{(\lambda_1+\lambda_2)(\lambda_1+\lambda_3)(\lambda_2+\lambda_3)}{\lambda_1\lambda_2\lambda_3(\lambda_1+\lambda_2+\lambda_3)}\right).$$

The existence was verified through order $q^7$ and later through order $q^{17}$, while uniqueness was checked through order $q^5$ in the cited work. The full conjecture remains open.