Multiplicity refinement for specialized spectral-invariant varieties

Let $V(I')$ be the affine variety defined by the specialized ideal $I'$ associated with the specialized spectral-invariant system, and let $q$ denote the period. A point is regular when it is nonsingular, and multiplicity refers to its scheme-theoretic multiplicity. Multiplicity refinement conjecture. If $q$ is odd, then $V(I')$ has only regular nonzero points. If $q$ is even and greater than $6$, then, up to symmetry, $V(I')$ has a single singular point of multiplicity $2^{\frac{q}{2}-2}$. Moreover, the multiplicity at the zero point is

$$2^{\left\lfloor \frac{q}{2}\right\rfloor}.$$

This refinement is based on computational data for the specialized variety and remains unresolved in the supplied text.