  We provide a proof of the following fact: if a complex scheme $Y$ has Behrend
function constantly equal to a sign $\sigma \in \{\pm 1\}$, then all of its
components $Z \subset Y$ are generically reduced and satisfy
$(-1)^{\mathrm{dim}_{\mathbb C} T_pY} = \sigma = (-1)^{\mathrm{dim}Z}$ for $p
\in Z$ a general point. Given the recent counterexamples to the parity
conjecture for the Hilbert scheme of points $\mathrm{Hilb}^n(\mathbb A^3)$, our
argument suggests a possible path to disprove the constancy of the Behrend
function of $\mathrm{Hilb}^n(\mathbb A^3)$.
