Extension of the lifted operator theorem to matrix Muckenhoupt weights

Let $1<p<\infty$ and let $W=(w_{ij})\colon \mathbb{R}^n\times \mathbb{R}^m\to\mathbb{C}^{N\times N}$ be a matrix weight in the set of Muckenhoupt weights $A_p(N,\mathbb{R}^n\times \mathbb{R}^m)$. Let $T_K$ be the operator lifted to the vector-valued setting, where the scalar kernel $K$ is of the type considered in Theorem

$$. **Matrix-weight extension conjecture.** The conclusion of Corollary$$

holds true for any such matrix weight $W$; in particular, $T_K$ is bounded on $L^p(\mathbb{R}^n\times\mathbb{R}^m,W)$. This would remove the technical obstacles to lifting the scalar result to the full matrix-weighted setting; the supplied text does not indicate whether the assertion has been proved or remains open.