Geramita's catalecticant ideal equality and inclusion conjecture

Let $K$ be a field, let $n,d,ke\mathbb{Z}$ with $k,n\geq 2$ and $d\geq 2k-2$, and let $Cat(t,d-t;n)$ denote the $t$-th generic catalecticant. Write $I_k(Cat(t,d-t;n))$ for the ideal generated by its $k\times k$ minors. Geramita's conjecture. For every $t$ with $k-1\leq t\leq d-k+1$, one has

$$I_k(Cat(k-1,d-k+1;n))=I_k(Cat(t,d-t;n)).$$

Moreover,

$$I_k(Cat(1,d-1;n))\subset I_k(Cat(2,d-2;n))\subset\cdots\subset I_k(Cat(k-1,d-k+1;n)).$$

This conjecture generalizes the questions about equality and inclusion of ideals of minors of catalecticants. The paper proves the corresponding case $k=3$, but the general assertion is presented as open.