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

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?