Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $ \DeclareMathOperator{\Aut}{Aut} \Aut(X)$.

Define a torus in $\Aut(X)$ to be a faithful algebraic action $α: (\mathbb{C}^*)^m \times X \to X$ of $T := (\mathbb{C}^*)^m$. It is maximal if there is no torus $T' \cong (\mathbb{C}^*)^{m+1}$ such that the action can be faithfully extended to $T'$.

Suppose $T_1, T_2$ are two maximal tori. Are they necessarily conjugate, i.e.\ does there exist a $γ ∈ \Aut(X)$ and an isomorphism of algebraic groups $ρ: T_1 \to T_2$ such that $γ ∘ α_1 ∘ γ^{-1} = α_2(ρ(-), -)$ ?

Most people seem to believe this is false. Are there any known counterexamples?