Infinite generation and odd-degree generators of the symplectic commutative graph algebra

Let $H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}$ be the symplectic-invariant homology algebra, let $\mathcal{A}^+(\phi)$ be the subalgebra generated by the positive-degree elements of $\mathcal{A}(\phi)$, and let

$$\mathcal{E}=H_*(\mathfrak{c}^+_\infty)^{\mathrm{Sp}}/\mathcal{I}(\mathcal{A}^+(\phi))$$

be the resulting free graded commutative algebra. The conjecture. The free graded algebra $\mathcal{E}$ is infinitely generated, and there exist infinitely many generators with odd degrees. The preceding results identify the initial low-weight pieces of $\mathcal{E}$, but do not determine its full generating set; the asserted infinite generation and odd-degree phenomenon remain open.