Zanello's Gorenstein Interval Conjecture

Let $(1,h_1,\ldots,h_e)$ be a Gorenstein $h$-vector, so $h_e=1$ and the entries are symmetric. Fix an index $i$ and a positive integer $\alpha$. Suppose that both

$$(1,h_1,\ldots,h_i,\ldots,h_{e-i},\ldots,h_{e-1},h_e=1)$$

and

$$(1,h_1,\ldots,h_i+\alpha,\ldots,h_{e-i}+\alpha,\ldots,h_{e-1},h_e=1)$$

are Gorenstein $h$-vectors. Gorenstein Interval Conjecture. Then

$$(1,h_1,\ldots,h_i+\beta,\ldots,h_{e-i}+\beta,\ldots,h_{e-1},h_e=1)$$

is also Gorenstein for every integer $\beta=0,1,\ldots,\alpha$. The conjecture is a symmetric generalization of the Interval Conjecture and proposes strong regularity for the set of Gorenstein Hilbert functions, whose complete description is difficult; it remains open in the source.