Classification conjecture for strongly-neutralizable threshold functions

A threshold function is a map $f:\Sigma^N\to\Sigma$, where $\Sigma$ is the binary state space. A function is strongly neutralizable when it has the property that its associated Ising machine can be made neutralizable under the paper's definition. An $n$-dimensional function is an extrusion of a lower-dimensional function if it ignores an added variable, namely $f^e(s_0,\ldots,s_n):=f(s_1,\ldots,s_n)$. Let $\mathcal G(N,M)$ denote the Ising symmetry group acting on threshold functions; write $\operatorname{AND}^{sd}$ for the self-dualization of the AND gate. Classification conjecture. Up to $\mathcal G(N,M)$ action, the only strongly-neutralizable threshold functions of dimension at least $2$ that are not extrusions of lower-dimensional functions are the AND gate and its self-dualization $\operatorname{AND}^{sd}$. This conjecture would classify the non-redundant strongly-neutralizable threshold functions. The authors report that they have checked the claim only through dimension $7$, so the classification remains open beyond that range.