Symmetric-group log-concavity conjecture for the ideals I(n) and J(n)

For each positive integer $n$, let $I(n)$ and $J(n)$ be the ideals in the polynomial ring ${\mathbb C}[{\mathbf x}_{\binom{[n]}{2}}]$ on variables indexed by the two-element subsets of $[n]$, with the quotient modules carrying the natural ${\mathfrak S}_n$-action. A graded ${\mathfrak S}_n$-module is ${\mathfrak S}_n$-log-concave when its graded pieces admit the required equivariant injections between adjacent tensor products. Symmetric-group log-concavity conjecture. For integers $n>0$, both

$${\mathbb C}\Big[{\mathbf{x}}_{\binom{[n]}{2}}\Big]\Big/I(n)$$

and

$${\mathbb C}\Big[{\mathbf{x}}_{\binom{[n]}{2}}\Big]\Big/J(n)$$

are ${\mathfrak S}_n$-log-concave. The conjecture was verified computationally for $n\le 20$, but no general proof is given.