A note on the classification of simple $S{L}_2({\bar{\mathbb{F}}}_p)$-modules admitting $\mathbf{T}$-stable lines in cross characteristic

Let $\mathbf{T}$ be the group of diagonal matrices in $S{L}_2({\bar{\mathbb{F}}}_p)$, where $p$ is a prime number. Let $\mathbb{k}$ be an algebraically closed field of characteristic not equal to $2$ and $p$. We classify all the irreducible $\mathbb{k}$-representations of $S{L}_2({\bar{\mathbb{F}}}_p)$ that admit $\mathbf{T}$-stable lines.