Nonexistence of even-dimensional degree-two hypersensitive functions

Let an $(n,d)$-hypersensitive function, or $(n,d)$-HSF, be an $(n,d)$-PTF $f$ whose dichromatic count satisfies $\mathbf{D}[f]>\mathbf{D}[f^*_{n,d}]$.

Even-$n$, degree-two consequence. For every even $n$, $(n,2)$-HSFs do not exist.

This is presented as a consequence that would follow from the limiting Gotsman–Linial conjecture, rather than as an independent conjecture with a separate name. It is therefore merged here as a restatement/consequence of that revised conjecture.