Hilbert function conjecture for general linear exact zero divisors

Let $R$ be a standard graded algebra, and let $d>0$. A pair $(\ell,Q)$ of exact zero divisors means that $\ell$ is a general linear form, $Q$ is an element of degree $d-1$, and the pair satisfies the exact zero-divisor condition. Hilbert function conjecture. If for general linear forms $\ell$ there exists $Q$ of degree $d-1$ such that $(\ell,Q)$ is a pair of exact zero divisors, then

$$\operatorname{dim}(R_d)=\operatorname{dim}(R_{d-1})-1.$$

This conjecture seeks a converse to the result that, when $R_{\ge 3}=0$, the existence of one pair of exact zero divisors implies that general linear forms are part of such pairs. It predicts a Yoshino-type necessary condition on the Hilbert function without assuming $R_{\ge 3}=0$; its resolution is not established in the supplied text.