Is it possible to detect the phase $\pi$ or 0 for the single qubit circuit X H P?

Question

I found an answer that shows how to detect the phase in cases like $0$, $\pi/8$, $\pi/2$, $\pi/4$ or $\pi$ for circuit to prepare state as H P, where P is a phase gate like $I$, $U1(\pi/8)$, $S$, $T$ or $Z$.

But in my case the circuit to prepare state is $X H P$, where $P$ is $X$ gate (conditionally phase $\pi$) or $ID$ (conditionally phase 0).

This circuit in Qasm with conditionally phase equal to $\pi$:

x q[0];
h q[0];
x q[0];

with conditionally phase equal to $0$:

x q[0];
h q[0];
id q[0];

Appending $H$ gate (as in the above answer) don't detect a difference for conditionally phase $\pi$ and phase $0$ (but does detect for phases $\pi/2$, $\pi/4$, $\pi/8$ if $P$ is $S$, $T$, $U1(\pi/8)$, respectively).

Is it possible to detect the conditionally phase $P$ $\pi$ or 0 for this circuit to prepare state?

Answer 1

I think unlike the relative phase in the answer you reference, it is a global phase in your case:

Your XHP-circuit where P=ID, prepares the state: [0.707+0j,-0.707+0j], where P=X, prepares the state: [-0.707+0j, 0.707+0j]. These states are differ by a global phase ${e}^{i\pi}=-1$.

But the global phase is undetectable $|ψ⟩:={e}^{iδ}|ψ⟩$, also see the answer.