Projective-dual linear-section Hodge conjecture

Let $\mathsf{X}\subset\mathbb{P}(\mathsf{V})$ be a smooth projective variety whose Hodge conjecture is true, and let $\mathsf{Y}\subset\mathbb{P}(\mathsf{V}^{\ast})$ be its projective dual. For a linear subspace $\mathsf{L}\subset\mathsf{V}^{\ast}$, define the linear sections

$$\mathsf{X}_{\mathsf{L}}=\mathsf{X}\cap\mathbb{P}(\mathsf{L}^{\perp}),\qquad \mathsf{Y}_{\mathsf{L}}=\mathsf{Y}\cap\mathbb{P}(\mathsf{L}).$$

Assume that both sections have expected dimension and are smooth. Projective-dual linear-section conjecture. The Hodge conjecture for $\mathsf{X}_{\mathsf{L}}$ is equivalent to the Hodge conjecture for $\mathsf{Y}_{\mathsf{L}}$. This predicts a Hodge-theoretic duality for corresponding smooth linear sections of projectively dual varieties, even in situations where homological projective duality is not available; the source presents it as motivated by noncommutative methods and examples.