On a conjecture on Hodge loci of linear combinations of linear subvarieties

For each $k \geq 5$ we give a counterexample to a conjecture of Movasati on the dimension of certain Hodge loci of cubic hypersurfaces in $\mathbf{P}^{2k+1}$ containing two $k$-planes intersecting in dimension $k-3$. We give similar examples for Hodge loci of cubic hypersurfaces in $\mathbf{P}^{2k+1}$ containing two $k$-planes intersecting in dimension $k-2$ and for quartic hypersurfaces in $\mathbf{P}^{2k+1}$ containing two $k$-planes intersecting in dimension $k-2$. Moreover, we present new evidence for Movasati's conjecture for the values of $k$ for which our type of counterexamples cannot exist, i.e., for $k=3,4$.