The many-variable containment conjecture for symmetric ideals

Fix $r,n\in\mathbf{N}$. For a subset $S\subset [r]=\{1,2,\ldots,r\}$, define

$$t_S=\sum_{i\in S}t_i,\qquad t_{S,0}=t_0+t_S,$$

and let $I_r^{(d)}\subset R=K[t_0,\ldots,t_r]$ be

$$I_r^{(d)}=\langle (t_{S,0})^d:S\subset [r]\rangle.$$

The many-variable containment conjecture. For any $r\geq 2$ and $n\geq 1$,

$$(t_1\cdots t_r)^{2n-1}\in I_r^{(nr)}.$$

For $r=1$, the analogous containment follows from the binomial expansion, and the theorem in the paper establishes the case $r=2$. The authors report computer verification for some small values of $r\geq3$, while the general assertion remains open.