The volume-invariant algebra conjecture for cyclic polytopes

Let $d$ and $n$ be positive integers, let $C^d_n$ be the cyclic $d$-polytope with $n$ vertices, and let $\mathsf{Inv}^d_{d+2}$ denote the algebra of degree-$d$ volume invariants for $C^d_{d+2}$. Let $\mathcal I(\mathsf{PL}^d_{d+2})$ be the kernel of $H^d_{d+2}$, and let $\mathsf{vol}_d$ be the signed volume invariant. Then the volume-invariant algebra conjecture asserts

$$\mathsf{Inv}^d_{d+2}/\mathcal I(\mathsf{PL}^d_{d+2}) \cong \mathbb R[H^d_{d+2}(\mathsf{vol}_d)]\subseteq \mathcal S^d[x_1,\dots,x_{d+2}],$$

i.e. the quotient is the subalgebra generated by $H^d_{d+2}(\mathsf{vol}_d)$. Equivalently,

$$\mathsf{Inv}^d_{d+2}=\operatorname{span}\{(\mathsf{vol}_d)^{\shuffle k}:k\geq 0\}\oplus \mathcal I(\mathsf{PL}_{d+2}^d).$$

The claim concerns the remaining difficult cases of the ring of volume invariants after the kernel is accounted for, and is based on computations in low dimensions. Its resolution is not supplied here.