Movasati's codimension conjecture for Hodge loci of two intersecting planes

Let $k\geq 2$ be an integer. Let $X$ be a hypersurface of degree $3$ in $\mathbf{P}^{2k+1}$ containing two $k$-planes $\Pi_1$ and $\Pi_2$ whose intersection has dimension $k-3$. For integers $a,b$ with $\gcd(a,b)=1$ and $ab\neq 0$, write $\operatorname{NL}(a[\Pi_1]+b[\Pi_2])$ for the Hodge locus where the corresponding linear combination remains of type $(k,k)$, and $\operatorname{NL}([\Pi_1],[\Pi_2])$ for the locus where both classes remain of type $(k,k)$. Movasati's conjecture.

$$\operatorname{codim} \operatorname{NL}(a[\Pi_1]+b[\Pi_2])=\operatorname{codim} \operatorname{NL}([\Pi_1],[\Pi_2])-1.$$

This conjecture concerns the irreducible components and codimensions of Hodge loci associated with linear combinations of the classes of two intersecting planes. The paper presents new evidence for it in the cases where the authors' counterexamples cannot occur, while the general claim is not resolved by the supplied context.