Flatness and Cohen–Macaulayness conjecture for deformed arc-space algebras

Let $\mathcal{C}$ be the cone in the source, let $\mathpzc{Arc}_q^N$ be the mapping space associated with the deformation $\mathbb{C}[z,q]/(z^{N+1}-q^{N+1})$, and set $J_q^N=\mathcal{O}[\mathpzc{Arc}_q^N]$. Arc-space deformation conjecture. The algebra $J_q^N$ is flat over $\mathbb{C}[q]$, and it is a Cohen–Macaulay integral domain over $\mathbb{C}[q]$. These properties would control the singularities and specialization of the deformed finite arc spaces, with the stated Hilbert-series consequences for the special fiber.