The Gorenstein Interval Conjecture for Gorenstein Hilbert functions

Let $A$ be an artinian standard graded algebra over a field $k$, and write its Hilbert function as $h=(1,h_1,\dots,h_e)$, where $e$ is the socle degree. A Hilbert function is Gorenstein when it is the Hilbert function of a Gorenstein algebra. Suppose that for some $\alpha\geq 2$, both

$$(1,h_1,\dots,h_i,\dots,h_{e-i},\dots,h_{e-1},1)$$

and

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

are Gorenstein Hilbert functions. Gorenstein Interval Conjecture. Then

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

is also a Gorenstein Hilbert function for every $\beta=1,2,\dots,\alpha-1$. The conjecture asserts that the Gorenstein Hilbert functions form an interval under the indicated simultaneous increase of the entries. The paper proves the conjecture in low socle degree, while the general statement remains the subject of the conjecture.