The $L_1$ Sobolev cubature lower-bound conjecture

Let $\mathbf W^r_{1,\alpha}$ be the weighted Sobolev-type function class, and let $\kappa_m(\mathbf W^r_{1,\alpha})$ denote the minimal cubature error over formulas with $m$ knots. $L_1$ Sobolev cubature lower-bound conjecture. For any $d\ge 2$ and any $r\ge 1$,

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

This is one of the two big open problems identified in the paper. A corresponding logarithmic lower bound is known for restricted weights and for the zero-weight parameter, but the unrestricted assertion remains open.