The codegree isomorphism conjecture for finite simple groups

Let $G$ be a finite group. For an irreducible character $\chi$ of $G$, define its codegree by

$$\operatorname{cod}(\chi)=\frac{|G:\operatorname{ker}(\chi)|}{\chi(1)}.$$

Let $\operatorname{cod}(G)$ denote the set of codegrees of irreducible characters of $G$.

The codegree isomorphism conjecture. Let $H$ be a finite nonabelian simple group and $G$ a finite group. Then

$$\operatorname{cod}(G)\subseteq\operatorname{cod}(H)\quad\text{if and only if}\quad G\cong H.$$

This is proposed as a stronger version of the codegree isomorphism conjecture, whose equality hypothesis has been confirmed for several families of simple groups. The supplied text does not give a resolution of the stronger version.