Brzozowki's algorithm doesn't work for this corner case I'm a newbee learning DFA minimization. And I found that(strangely) Brzozowki's algorithm cannot give me a minimized DFA on this example:

In this DFA, $S_0$ and $S_1$ are nondistinguishable and should be merged after minimization, but they are not.
The process of minimization is like this:

I tried to find out why by reading @Hendrik Jan's proof here, Proof of Brzozowski&#39;s algorithm for DFA minimization?

However, I find that this case bypasses the proof, because that in the reversed DFA, the initial state $q_0$ cannot be reached by $q_0$ using any words (but initial state is always considered reachable and cannot be removed). So did I miss anything, or does Brzozowki's algorithm has any specification for this corner case?

(It bypasses the proof like this: in the second series of reversal-determinization, $S_0 = \{q_1\}$ and $S_1 = \{q_0, q_1\}$, for any $w$, $\delta(q_0, w)$ in $S_0$ iff $\delta(q_0, w)$ in $S1$, so we say $S_0 
= S_1$, assuming all states can be reached by $\delta(q_0, w)$, but $\delta(q_0,w)$ will never be $q_0$)

the initial DFA I was working on:
I have looked into some other corner cases and have a tentative answer: when we are doing the determinization after reversal, there is a corner case where in the DFA before reversal, some states cannot be reached  by applying any word to the initial state (two possibilities: the initial state cannot go back to itself or we are adding a state as the new initial state, because when we are doing reversal, there are 2 or more final states). In these cases, two sets are considered the same set if the only differences between them are the "unreachable" states before reversal.

I think we can prove that by reversing the proof process in the link cited in the question.