  In this note we establish a (non-trivial) lower bound on the degree two entry
$h_2$ of a Gorenstein $h$-vector of any given socle degree $e$ and any
codimension $r$.
  In particular, when $e=4$, that is for Gorenstein $h$-vectors of the form
$h=(1,r,h_2,r,1)$, our lower bound allows us to prove a conjecture of Stanley
on the order of magnitude of the minimum value, say $f(r)$, that $h_2$ may
assume. In fact, we show that $$\lim_{r\to \infty} {f(r)\over r^{2/3}}=
6^{2/3}.$$
  In general, we wonder whether our lower bound is sharp for all integers
$e\geq 4$ and $r\geq 2$.
