Population of excited H levels in a Strömgren Sphere
There just aren't any other mechanisms to keep the population excited in a Stromgren sphere.  In principle collisions can also lead to excitations, but if you do an order of magnitude calculation you'll find that the time between collisions is much longer than the time to receive an ionizing photon.  In a laboratory setting you might be able to coerce some particles into remaining in an excited state using the Quantum Zeno effect, but that won't happen in interstellar space.

So without something to keep the atoms in an excited state, even if you start with everything in the excited state, in time $\tau$ you will find that everything has decayed to the ground state, with the exception of some very small fraction $\tau_{\textrm{ion}} / \tau$, which may have moved up to a higher energy state.  Assuming the lifetimes of those very excited states are comparable, after a few more times $\tau$, you will find that even those will have decayed to the ground state.