Hodge Structures in Sextic Fourfolds Equipped with an Involution

To each ternary sextic $f(X_0, X_1, X_2)$ whose associated plane curve is smooth, the Shioda construction attaches a smooth sextic fourfold $X \subset \mathbb{P}^5$ whose defining equation $f(X_0, X_1, X_2) - f(Y_0, Y_1, Y_2)$ is fixed under the involution $\iota : (X_0, X_1, X_2, Y_0, Y_1, Y_2) \mapsto i \cdot (Y_0, Y_1, Y_2, -X_0, -X_1, -X_2)$. The induced action $\iota^* : H^4(X, \mathbb{Q}) \to H^4(X, \mathbb{Q})$ fixes a Hodge substructure $H \subset H^4(X, \mathbb{Q})$ whose Hodge coniveau is 1. By the general Hodge conjecture, we expect that there should exist a divisor $Y \subset X$ for which $H \subset \ker\left( H^4(X, \mathbb{Q}) \to H^4(X \setminus Y, \mathbb{Q}) \right)$. We verify this prediction in case the Waring rank of $f(X_0, X_1, X_2)$ takes on its minimum possible value, partially answering a question of Voisin (J. Math. Sci. Univ. Tokyo '15).