Symmetric-group log-concavity conjecture for orbit harmonics of bounded-block partitions

For integers $n$ and $m$, let $Pi_{n,m}$ be the locus of unordered set partitions of $[n]$ with block sizes bounded by $m$, and let $R( Pi_{n,m})$ be its orbit-harmonics graded ${\mathfrak S}_n$-module. Symmetric-group \log-concavity conjecture. For $n\ge 9$, the module

$$R(\Pi_{n,m})$$

is ${\mathfrak S}_n$-log-concave. The text notes that log-concavity fails for $n=7,8$ and $4\le m\le n$, while computational evidence supports the conjecture for $9\le n\le16$; the general assertion remains open.