# Is every paracompact topology contained in a maximal paracompact topology?

If $$(X,\tau)$$ is a paracompact, is there a topology $$\tau'\supseteq \tau$$ such that $$(X,\tau')$$ is still paracompact, and $$\tau'$$ is maximal with respect to $$\subseteq$$ and paracompactness?

Any discrete space is paracompact, since the family of singletons is locally finite and an open refinement of every open cover. Put $$\tau'=\mathcal{P}(X)$$. Then the discrete space $$(X,\tau')$$ is paracompact, and $$\tau'$$ is maximal with respect to inclusion and paracompactness.