What would $P = NP$ tell us about how the polynomial hierarchy relates to $PSPACE$

Question

If $P = NP$ it's easy to show by induction that the $PH$ collapses to $P$. But would that have any other implications for the relation between $PSPACE$ and $PH$? We already know that $PH \subseteq PSPACE$ and this continues to hold true in the case of the $PH$ collapsing, but would $P = NP$ imply anything else for the relation(for example, $PSPACE \subseteq PH$?)