The $L_\infty$ Sobolev cubature lower-bound conjecture

Let $\mathbf W^r_{\infty,\alpha}$ be the weighted Sobolev-type function class, and let $\kappa_m(\mathbf W^r_{\infty,\alpha})$ denote the minimal cubature error over formulas with $m$ knots. $L_\infty$ Sobolev cubature lower-bound conjecture. For any $d\ge 2$ and any $r>0$,

$$\kappa_m(\mathbf W^r_{\infty,\alpha}) \ge C(r,d)m^{-r}(\log m)^{(d-1)/2}.$$

This is the second big open problem identified in the paper. General nontrivial lower estimates for the $p=\infty$ case are unavailable, although the paper gives a conditional result; consequently this conjecture remains open.