Matrix weighted square function conjecture

Let $W$ be a $d\times d$ matrix weight in $A_2$, and let $S_W$ denote the associated square function on $L^2(W)$. Matrix weighted square function conjecture.

$$\|S_W f\|_{L^2(\mathbb{R},\mathbb{R})}^2\lesssim [W]_{A_2}^2\|f\|_{L^2(W)}^2\qquad\forall f\in L^2(W).$$

The scalar analogue holds with the sharper dependence $[w]_{A_2}^2$ rather than $[w]_{A_2}^2\log [w]_{A_2}$; whether the logarithmic factor can be removed for matrix weights is left open.