Bernshteyn–Lee's lower-bound conjecture for weak degeneracy of regular graphs

A graph $G$ is weakly $d$-degenerate if it is weakly $f$-degenerate for the constant function with value $d$, and its weak degeneracy $\operatorname{wd}(G)$ is the minimum such integer $d$. Bernshteyn–Lee's conjecture. Every $d$-regular graph $G$ satisfies

$$\operatorname{wd}(G)\geqslant d-O(\sqrt{d}).$$

The conjecture would improve the general lower bound $\operatorname{wd}(G)\geqslant d-\sqrt{2n}$ for a $d$-regular graph with $n\geqslant2$ vertices, showing that weak degeneracy is close to the degree independently of the graph's order. The trivial upper bound is $d$, attained by complete graphs.