Conjecture on smooth and oversized Hodge loci for cubic hypersurfaces

Let $X_0$ be a cubic hypersurface in the full family of cubic hypersurfaces, and let $\mathbb{P}^{\frac{n}{2}}$ and $\check{\mathbb{P}}^{\frac{n}{2}}$ be linear cycles in $X_0$. For a Hodge locus $V_\gamma$, write $V_{[\mathbb{P}^{\frac{n}{2}}]-[\check{\mathbb{P}}^{\frac{n}{2}}]}$ and $V_{r[\mathbb{P}^{\frac{n}{2}}]-\check r[\check{\mathbb{P}}^{\frac{n}{2}}]}$ for the loci associated with the indicated cohomology classes. Smooth and oversized Hodge-locus conjecture. For the full family of cubic hypersurfaces, the following hold: (1) if $\mathbb{P}^{\frac{n}{2}}\cap\check{\mathbb{P}}^{\frac{n}{2}}=\mathbb{P}^{\frac{n}{2}-2}$, then $V_{[\mathbb{P}^{\frac{n}{2}}]-[\check{\mathbb{P}}^{\frac{n}{2}}]}$ is smooth and larger than the deformation space of the triple $(X_0,\mathbb{P}^{\frac{n}{2}},\check{\mathbb{P}}^{\frac{n}{2}})$; (2) if $\mathbb{P}^{\frac{n}{2}}\cap\check{\mathbb{P}}^{\frac{n}{2}}=\mathbb{P}^{\frac{n}{2}-3}$ and $|r|,|\check r|\in\mathbb{N}$, then $V_{r[\mathbb{P}^{\frac{n}{2}}]-\check r[\check{\mathbb{P}}^{\frac{n}{2}}]}$ is smooth and larger than the deformation space of the same triple, with dimension difference $1$. The conjecture concerns excess-dimensional Hodge loci in the full cubic family; the supplied status evidence says that the integral and rational Hodge conjecture are proved in this case, so this candidate is recorded as solved.