Maximal subgroups acting with two composition factors on irreducible representations of exceptional algebraic groups

Let $Y$ be a simply connected simple algebraic group over an algebraically closed field $k$ of characteristic $p$ and let $X$ be a maximal closed connected simple subgroup of $Y$. Excluding some small primes in specific cases, we classify the $p$-restricted irreducible representations of $Y$ on which $X$ acts with exactly two composition factors. This work follows on naturally from the classification of irreducible subgroups of exceptional algebraic groups given by Testerman.