Off-diagonal bump conjecture for sparse operators

Let $T$ be a sparse operator, let $1<p<q<\infty$, and let $\mathcal{D}_{p,p}$ and $\mathcal{D}^{\ast}_{p,p}$ denote the corresponding direct comparison bump quantities for the case $p=q$. Off-diagonal bump conjecture. One has

$$\left\|T\sigma\cdot:L^p(\sigma)\to L^p(w)\right\| \lesssim \mathcal{D}_{p,p}+\mathcal{D}^{\ast}_{p,p}.$$

This conjecture concerns two-weight boundedness for sparse operators and is motivated by comparisons with Orlicz bump conjectures. The source notes that Lerner almost proves it, but does not state that it has been resolved.