Non-unimodal Artinian Gorenstein algebras in embedding dimension 4

The problem asks whether a standard graded Artinian Gorenstein algebra of embedding dimension $4$ can have a non-unimodal Hilbert function. The literature records unimodality in dimensions at most $3$ and examples in every dimension at least $5$, leaving dimension $4$ unresolved.

Known results

• Stanley (1978): the smallest known non-unimodal example is $(1,13,12,13,1)$, in codimension $13$.
• The 2007 analysis proves unimodality in codimension $4$ when $h_4\le 33$; its remaining case is $h_4=34$.
• The 2010 paper proves unimodality in codimension $4$ if a minimal generator has degree less than $5$, and also when $h_4\le 34$.
• The 2022 work reiterates that examples exist in every codimension $r\ge 5$, while codimension $4$ remains open.

Current status (as of August 2026): Unimodality is established for several substantial codimension-$4$ subclasses, but the existence of a non-unimodal example in embedding dimension $4$ remains open.