Domingo-Salazar–Lacey–Rey weighted weak-type conjecture for the intrinsic square function

Let $S_{\operatorname{int}}$ be the intrinsic square function and let $w\in A_2$ be a Muckenhoupt weight, with characteristics $[w]_{A_2}$ and $[w]_{A_{\infty}}$. The weighted weak space is denoted by $L^{2,\infty}(w)$. Domingo-Salazar–Lacey–Rey conjecture. For $w\in A_2$,

$$\lVert S_{\operatorname{int}}f \rVert_{L^{2,\infty}(w)} \lesssim [w]_{A_2}^{1/2}\left(1+\log_{+}[w]_{A_{\infty}}\right)^{1/2}\lVert f\rVert_{L^2(w)},$$

and this estimate is sharp. This is the critical weighted weak-$L^2$ estimate for the intrinsic square function; the source presents it as a conjecture and gives no resolution evidence.