Adjoint Steinberg conjecture

Let $\tilde{G}$ be the relevant reductive group, let $Z(\tilde{G})$ denote its center, and let $V(\lambda_k)$, for $k=1,\ldots,l$, be the fundamental representations under consideration. Two semisimple elements are considered equivalent in each setting up to multiplication by a scalar, and in $\tilde{G}$ up to multiplication by an element of $Z(\tilde{G})$. Adjoint Steinberg conjecture. Semisimple elements of $\tilde{G}$ are conjugate in $\tilde{G}$ up to a multiple of an element in $Z(\tilde{G})$ if and only if they are conjugate in $\operatorname{GL}(V(\lambda_k))$ up to a constant for $k=1,\ldots,l$. This question is motivated by Steinberg's related conjecture concerning detection of conjugacy by all irreducible rational representations; the claim is known for groups of $A_l$-type in the cases described in the source, but remains open in general.