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 ${\iota }^{\ast }:H^4(X,\mathbb{Q})\to H^4(X,\mathbb{Q})$ 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 $H\subset \ker (H^4(X,\mathbb{Q})\to H^4(X\setminus Y,\mathbb{Q}))$. 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).