Quantum cohomology semisimplicity conjecture for linear sections of (co)adjoint varieties

Let $X$ be an adjoint or quasi-minuscule variety as above, and let $Y\subset X$ be a general linear section. Write ${\rm BQH}(Y)$ and ${\rm QH}(Y)$ for its small and big quantum cohomology algebras. Semisimplicity conjecture. The algebra ${\rm BQH}(Y)$ is semi-simple, and ${\rm QH}(Y)$ is semi-simple if and only if ${\rm QH}(X)$ is semi-simple. This extends the known semisimplicity results for the ambient adjoint and quasi-minuscule varieties to their general linear sections; the paper proves partial results, but the full statement remains open.