Codegrees conjecture for finite nonabelian simple groups

Let $H$ be a finite nonabelian simple group, and let $G$ be a finite group. Write ${\mathrm {cod}}(G)$ for the set of character codegrees of $G$. Codegrees conjecture. If

$${\mathrm {cod}}(G)={\mathrm {cod}}(H),$$

then $G\cong H$. This is the codegree analogue of Huppert's character-degree conjecture. The source presents it as a problem motivated by the degree conjecture; no resolution is supplied in the stated passage.