  Let $S$ be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In
this short note, we propose the conjecture that the Betti poset of $I$
determines the Stanley projective dimension of $S/I$ or $I$. Our main result is
that this conjecture implies the Stanley conjecture for $I$, and it also
implies that \[ \operatorname{sdepth} S/I \geq \operatorname{depth} S/I - 1.\]
Recently, Duval et al. found a counterexample to the Stanley conjecture, and
their counterexample satisfies $\operatorname{sdepth} S/I =
\operatorname{depth} S/I - 1$. So if our conjecture is true, then the
conclusion is best possible.
