  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 $k=3,4$.
