The Betti-number conjecture for elementary and complete homogeneous symmetric cubics

Let $n\geq 3$, and consider the ideals

$$I_h=(x_1^3,\ldots,x_n^3,h_3),\qquad I_e=(x_1^3,\ldots,x_n^3,e_3)$$

in $ck[x_1,\ldots,x_n]$, where $h_3$ and $e_3$ are respectively the complete homogeneous and elementary symmetric polynomials of degree $3$.

Betti-number conjecture. The ideals $I_h$ and $I_e$ have the same Betti numbers if and only if $n=3a$ for some integer $a$.

The conjecture concerns finer homological data than the Hilbert series. The source notes that agreement of Hilbert series is expected when $n$ is a multiple of $3$, while agreement of Betti tables is conjectured exactly in those cases; this remains open.