Canonical quantum cohomology conjecture for hyperplane sections of type $A_n$ adjoint varieties

Benedetti and Perrin posed the conjecture in 2022: for a type-$A_n$ adjoint variety and a general hyperplane section $Y$, the canonical algebra ${\rm QH}(Y)_{\rm can}$ is semisimple exactly for even $n$. They also established the even case; the ambient algebra ${\rm QH}(X)_{\rm can}$ has the same parity criterion.

Known results

• Benedetti–Perrin, 2022: ${\rm QH}(X)_{\rm can}$ is semisimple exactly when $n$ is even.
• Benedetti–Perrin, 2022: the hyperplane-section conjecture is proved for even $n$.

2025 related developments

A later Benedetti–Perrin paper gives criteria for generic semisimplicity of ordinary small quantum cohomology of several Grassmannian hyperplane sections, but it does not explicitly settle the canonical specialization ${\rm QH}(Y)_{\rm can}$ for type-$A_n$ adjoint varieties or the odd case.

Current status (as of August 2026): the conjecture is settled for even $n$, while the odd-$n$ case remains open with no verified proof or counterexample reported in the retrieved sources.