Asymptotic unimodality conjecture for level Hilbert functions

For positive integers $e$ and $t$, consider level Hilbert functions of codimension $r$, socle degree $e$, and type $t$. Asymptotic unimodality conjecture. For fixed positive integers $e$ and $t$, almost all codimension $r$ level Hilbert functions of socle degree $e$ and type $t$ are unimodal as $r$ tends to infinity. The source motivates this through asymptotic counting results and known unimodality of almost all pure $O$-sequences; its status is not resolved in the supplied text.