Kourovka Notebook Problem 21.135

For a finite group $G$, let $\chi_1(G)$ denote the totality of the degrees of all irreducible complex characters of $G$ with allowance for their multiplicities. Suppose that $H$ is a finite group with $\chi_1(H)=\chi_1(G)$. If $G$ has trivial solvable radical, must $H$ also have trivial solvable radical?