D8-brane partition-function DT/PT identity

Let $n,m$ be nonnegative integers indexing D8/D8$'$ and opposite-brane pairs, and let $v_\alpha=e^{\mathfrak{a}_\alpha}$, $w_\alpha=e^{\mathfrak{b}_\alpha}$, $x_\beta=e^{\mathfrak{c}_\beta}$, and $y_\beta=e^{\mathfrak{d}_\beta}$. Denote by $\mathcal{Z}^{\mathrm{D8},+}_{n|m}$ and $\mathcal{Z}^{\mathrm{D8},-}_{n|m}$ the partition functions defined using the two Jeffrey–Kirwan reference vectors. D8-brane DT/PT identity. One has

$$\mathcal{Z}^{\mathrm{D8},+}_{n|m}[\mathfrak{q},q_{1,2,3,4}]=\operatorname{MF}\left[\prod_{\alpha=1}^{n}\frac{w_{\alpha}}{v_{\alpha}}\prod_{\beta=1}^{m}\frac{y_{\beta}}{x_{\beta}}\right]\mathcal{Z}^{\mathrm{D8},-}_{n|m}[\mathfrak{q},q_{1,2,3,4}].$$

The source presents this as an interesting DT/PT-type correspondence between the two partition functions, but supplies no proof or resolution.