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

Let $χ$ be an idele class character over a number field $F$, and let $π,π'$ be any two cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$. We prove that the Rankin-Selberg $L$-function $L(s,\mathrm{Sym}^2(π)\times(\mathrm{Sym}^2 (π')\otimesχ))$ 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 $π$ is non-dihedral and $π'$ is not a twist of $π$. Until now, this was only known when $π=π'$, $π$ is self-dual, and $χ$ is trivial.