Hytönen–Lacey one-supremum conjecture for Calderón–Zygmund operators

Let $T$ be a Calderón–Zygmund operator, let $w$ and $\sigma$ be weights, and let $1<p<\infty$. For a cube $Q$, write $\langle w\rangle_Q$ and $\langle \sigma\rangle_Q$ for the averages over $Q$, and define, for any weight $v$,

$$A_\infty(v,Q):= \frac{1}{v(Q)}\int_Q M(v\chi_Q)(x)\\,dx.$$

Hytönen–Lacey one-supremum conjecture. The estimate

$$\\|T(\cdot\sigma)\\|_{L^p(\sigma)\rightarrow L^p(w)}\lesssim \sup_Q \langle w\rangle_Q^{\frac 1p}\langle \sigma\rangle_Q^{\frac 1{p'}}\left(A_\infty(w,Q)^{\frac 1{p'}}+A_\infty(\sigma,Q)^{\frac 1p}\right)$$

should hold. This is a proposed one-supremum strengthening of mixed $A_p$–$A_\infty$ bounds; the supplied text gives no resolution status.