Efficient computation of the degree of a neural ideal

Let $d\ge 2$ and let $\mathcal C\subseteq 2^{[n]}$ be a code. A pseudo-monomial is an element of the canonical form $\operatorname{CF}(J_\mathcal C)$ of the neural ideal $J_\mathcal C$, and its degree is the number of factors in the pseudo-monomial.

Efficient degree computation conjecture. There is a polynomial-time algorithm that, given the codewords of $\mathcal C$ as binary vectors of length $n$, determines whether every pseudo-monomial in $\operatorname{CF}(J_\mathcal C)$ has degree at most $d$.

Efficiently computing the canonical form is relevant to the proposed algorithm for recognizing inductively pierced codes, since the algorithm begins by computing $\operatorname{CF}(J_\mathcal C)$. The supplied statement asserts the algorithm's existence but gives no resolution, so its status remains open.