The DT–PT₀ vertex correspondence without PT₀ moduli

Let $\boldsymbol\mu=\{\mu_a\}_{a=1}^{4}$ be plane partitions with no PT$_0$ moduli, and let $\mathsf V^{\mathrm{DT}}_{\mu_1\mu_2\mu_3\mu_4}(q)$ and $\mathsf V^{\mathrm{PT}_0}_{\mu_1\mu_2\mu_3\mu_4}(q)$ be the associated vertex contributions. DT–PT₀ vertex correspondence. There exists a choice of $\pm\sqrt{\cdot}$ at the $(\mathbb C^*)^4$-fixed points such that

$$\frac{\mathsf V^{\mathrm{DT}}_{\mu_1\mu_2\mu_3\mu_4}(q)}{\mathsf V^{\mathrm{DT}}_{\varnothing\varnothing\varnothing\varnothing}(q)}=\mathsf V^{\mathrm{PT}_0}_{\mu_1\mu_2\mu_3\mu_4}(q).$$

The accompanying remark makes the additional expectation that the relevant PT$_0$ vertices have no positive $T$-fixed term. This is a local toric vertex form of the DT–PT₀ correspondence, and no resolution is stated.