Conjecture on minimal generators of the annihilator of the complete symmetric form

Let $R$ be the polynomial ring and let $H_{2d+1}$ be the complete symmetric form considered in the source. Write $\operatorname{Ann}(H_{2d+1})$ for its annihilator ideal, and let $\varphi(F_{\underline a})$ denote the indicated elements from the source, with $|\underline a|$ denoting the sum of the entries of $\underline a$.

Minimal-generator conjecture. If $d$ is odd, then $\operatorname{Ann}(H_{2d+1})$ has no minimal generators in degree $d+2$, and

$$\operatorname{Ann}(H_{2d+1})=(\varphi(F_{\underline a}):|\underline a|=d+1).$$

If $d$ is even, then $\operatorname{Ann}(H_{2d+1})$ has exactly one minimal generator in degree $d+2$, and

$$\operatorname{Ann}(H_{2d+1})=(\varphi(F_{\underline a}):|\underline a|=d+1)+(\varphi(x_n^2(x_{n-1}-x_n)^d)).$$

The conjecture is motivated by computational experiments on determining a minimal generating set for the annihilator of the complete symmetric form. The supplied text gives no evidence of resolution.