  In 1991, Craig Gotsman and Nathan Linial conjectured that for all $n$ and
$d$, the average sensitivity of a degree-$d$ polynomial threshold function on
$n$ variables is maximized by the degree-$d$ symmetric polynomial which
computes the parity function on the $d$ layers of the hypercube with Hamming
weight closest to $n/2$. We refute the conjecture for almost all $d$ and for
almost all $n$, and we confirm the conjecture in many of the remaining cases.
