Bloom type upper bounds in the product BMO setting

For a bounded singular integral $T_n$ in ${\mathbb{R}}^n$ and a bounded singular integral $T_m$ in ${\mathbb{R}}^m$ we prove that $$\mathrm{\parallel }[T_n^1,[b,T_m^2]]{\mathrm{\parallel }}_{L^p(\mu )\to L^p(\lambda )}{\lesssim }_{[\mu {]}_{A_p},[\lambda {]}_{A_p}}\mathrm{\parallel }b{\mathrm{\parallel }}_{{\mathrm{BMO}}_{\text{prod}}(\nu )},$$ where $p\in (1,\mathrm{\infty })$, $\mu ,\lambda \in A_p$ and $\nu :={\mu }^{1\mathrm{/}p}{\lambda }^{-1\mathrm{/}p}$. Here $T_n^1$ is $T_n$ acting on the first variable, $T_m^2$ is $T_m$ acting on the second variable, $A_p$ stands for the bi-parameter weights of $$\mathbb{R}$^n \times $\mathbb{R}$^m$and$\operatorname{BMO}_{$\text{prod}$}(\nu)$ is a weighted product BMO space.