The tight log-rank conjecture for AND-functions

Let $f: \{0,1\}^n \to \{0,1\}$ be a Boolean function, and let $\operatorname{spar}(f)=r$ be its sparsity, namely the number of nonzero coefficients in its unique multilinear polynomial over the reals. Let $f_{\land}(x,y)=f(x_1y_1,\ldots,x_ny_n)$, and write $\mathrm{P}^{\land\text{-dt}}(f)$ for deterministic AND-decision-tree depth. Tight log-rank conjecture. One should have

$$\mathrm{P}^{\land\text{-dt}}(f)\leq \operatorname{poly}(\log r).$$

The conjecture would remove the extraneous $\log n$ factor from the paper's deterministic log-rank theorem and would also imply a tighter lifting theorem for AND-functions. The source gives no resolution evidence.