  Let $G$ be a finite group and let $cd(G)$ be the set of all irreducible
complex character degrees of $G$. It was conjectured by Huppert in Illinois J.
Math. 44 (2000) that, for every non-abelian finite simple group $H$, if
$cd(G)=cd(H)$ then $G\cong H\times A$ for some abelian group $A$. In this
paper, we confirm the conjecture for the family of projective special linear
groups $\textrm{PSL}_4(q)$ with $q\geq 13$.
