Exceptional zeros of ${\mathrm{G}\mathrm{L}}_3\times {\mathrm{G}\mathrm{L}}_3$ Rankin-Selberg $L$-functions

Let $\chi$ be an idele class character over a number field $F$, and let $\pi ,{\pi }^{\mathrm{\prime }}$ be any two cuspidal automorphic representations of ${\mathrm{G}\mathrm{L}}_2({\mathbb{A}}_F)$. We prove that the Rankin-Selberg $L$-function $L(s,{\mathrm{S}\mathrm{y}\mathrm{m}}^2(\pi )\times ({\mathrm{S}\mathrm{y}\mathrm{m}}^2({\pi }^{\mathrm{\prime }})\otimes \chi ))$ has a “standard” zero-free region with no exceptional Landau-Siegel zero except possibly when it is divisible by the $L$-function of a real idele class character. In particular, no such zero exists if $\pi$ is non-dihedral and ${\pi }^{\mathrm{\prime }}$ is not a twist of $\pi$. Until now, this was only known when $\pi ={\pi }^{\mathrm{\prime }}$, $\pi$ is self-dual, and $\chi$ is trivial.