The combinatorial conjecture for the refined local Donaldson–Thomas partition function

Let $\lambda$ be a Young diagram. For each box $\Box\in\mathbb{Z}^2_{\geq 0}$, let $p_{\lambda,\Box}=\prod_{\Box'\in H_\lambda(\Box)}q_{\Box'}$, where $H_\lambda(\Box)$ is the hook at $\Box$ with respect to $\lambda$. Let $\mathsf{Z}^{\mathrm{spp}}_{\lambda}(\mathbf{q})$ denote the refined local Donaldson–Thomas partition function associated with $\lambda$. The combinatorial conjecture. The following identities hold:

$$\mathsf{Z}^{\mathrm{spp}}_{\lambda}(\mathbf{q})=\prod_{\Box\in\mathbb{Z}^2_{\geq 0}\setminus\lambda}\frac{1}{1-p_{\lambda,\Box}}$$ $$=\prod_{\Box\in\mathbb{Z}^2_{\geq 0}}\frac{1}{1-p_{\varnothing,\Box}}\cdot\prod_{\Box\in\lambda}\frac{1}{1-p_{\lambda,\Box}}.$$

This is a conjectural product formula motivated by the corresponding discussion of motives of symmetric plane partitions. The supplied passage does not provide evidence resolving the conjecture.