The $h_2=r$ conjecture for artinian Gorenstein algebras

Let $R/I$ be an artinian Gorenstein algebra of codimension $r\geq 3$ and socle degree $e$, and assume that $h_2=r$. The $h_2=r$ conjecture. Then

$$h_i=r\qquad\text{for }i=1,2,\dots,e-1.$$

Furthermore, if $e\geq 4$, the ideal $(I)_{\leq e-1}$ generated by all forms in $I$ of degree at most $e-1$ is the saturated ideal of a zero-dimensional scheme, and $I$ has an additional $r-1$ minimal generators in degree $e$. The claim is presented as the paper's final conjecture and is not resolved there.