The uniqueness conjecture for symmetric decompositions of 2-admissible I-Gorenstein sequences

Let $A=k[\![x,y]\!]/J$ be an AG $k$-algebra, let $I$ be an ideal in $A$, and let $H$ be a $2$-admissible $G(I)$-Gorenstein sequence. A symmetric decomposition is a decomposition of the Hilbert function $H$ into symmetric component sequences. The uniqueness conjecture. The symmetric decomposition of $H$ is unique. Symmetric decomposition is known to be determined by the Hilbert function in the codimension-two AG case, but the examples in the paper show that a general $2$-admissible $I$-Gorenstein sequence need not determine its decomposition; the conjecture isolates the case asserted in the source and remains open.