Almost cyclic regular elements in irreducible representations of simple algebraic groups

Let $G$ be a simple linear algebraic group defined over an algebraically closed field of characteristic $p\geq 0$ and let $\phi$ be a $p$-restricted irreducible representation of $G$. Let $T$ be a maximal torus of $G$ and $s\in T$. We say that $s$ is strongly regular if $\alpha(s)\ne\beta(s)$ for all distinct $T$-roots $\alpha$ and $\beta$ of $G$. Our main result states that if all but one of the eigenvalues of $\phi(s)$ are of multiplicity 1 then, with a few specified exceptions, $s$ is strongly regular. This can be viewed as an extension of our earlier result saying that under the same hypotheses, $s$ must be regular and all non-zero weights of $\phi$ are of multiplicity 1.