Conjecture on socle elements in codimension-3 graded algebras

Let ${\bf H}$ be a Hilbert function of the form

$$\begin{matrix} {\bf H} &: & h_0 & h_1 & \cdots & h_{d-1} & h_d & h_d & h_{d+2} & \cdots \end{matrix}$$

with $h_d<h_{d+2}$. Socle-element conjecture. (a) Any graded algebra of codimension $3$ with Hilbert function ${\bf H}$ has a socle element in degree $d$. (b) If a graded algebra of codimension $3$ with Hilbert function ${\bf H}$ has a socle element in degree $d$, then some algebra with Hilbert function

$$\begin{matrix} {\bf H} &: & h_0 & h_1 & \cdots & h_{d-1} & \underbrace{h_d\ \cdots\ h_d}_{s\text{-times}} & h_{d+s} & \cdots \end{matrix}$$

with $s\geq 2$ has a socle element in degree $d+s-2$. This conjecture concerns the possible socle degrees of graded algebras with a fixed Hilbert function in codimension $3$; the source gives no evidence of a resolution, so its status remains open.