The non-smoothness conjecture for cubic Hodge loci

Let $V_{[Z]}$ be the Hodge locus associated with $Z=r\mathbb{P}^{\frac n2}+\check r\check{\mathbb{P}}^{\frac n2}$ on a smooth cubic hypersurface, where the two linear cycles meet in $\mathbb{P}^m$. Cubic non-smoothness conjecture. If $(d,m)=(3,\frac n2-2)$ and $(r,\check r)\ne(1,-1)$, then $V_{[Z]}$ is not smooth. The supplied text says this is proved for $n=6,8$ and for $(n,d,m)=(4,4,0)$ in the cited work, leaving the general assertion unresolved here.