Containment conjecture for initial annihilator ideals of generic forms

Let $R$ and $S$ be the polynomial rings considered in the source. Let $G\in S$ be a generic form of degree $e\geq 1$, and set

$$d=\lfloor e/2\rfloor+1.$$

Let $I$ be an ideal of $R$ generated by generic forms of degrees at least $d$, with at least $\dim_k R_d-\dim_k R_{d-1}$ of those generators having degree $d$.

Initial-ideal containment conjecture. Then

$$\operatorname{in}(\operatorname{Ann}(G))\subseteq \operatorname{in}(I).$$

This prediction arises from computational evidence comparing the initial ideal of the annihilator of a generic form with initial ideals of ideals generated by sufficiently many generic forms. The supplied text gives no evidence of resolution.