  Roughly ten years ago, the following "Gorenstein Interval Conjecture" (GIC)
was proposed: Whenever $(1,h_1,\dots,h_i,\dots,h_{e-i},\dots,h_{e-1},1)$ and
$(1,h_1,\dots,h_i+\alpha,\dots,h_{e-i}+\alpha,\dots,h_{e-1},1)$ are both
Gorenstein Hilbert functions for some $\alpha \geq 2$, then
$(1,h_1,\dots,h_i+\beta,\dots,h_{e-i}+\beta,\dots,h_{e-1},1)$ is also
Gorenstein, for all $\beta =1,2,\dots,\alpha -1$. Since an explicit
characterization of which Hilbert functions are Gorenstein is widely believed
to be hopeless, the GIC, if true, would at least provide the existence of a
strong, and very natural, structural property for such basic functions in
commutative algebra. Before now, very little progress was made on the GIC.
  The main goal of this note is to prove the case $e\le 5$, in arbitrary
codimension. Our arguments will be in part constructive, and will combine
several different tools of commutative algebra and classical algebraic
geometry.
