Question: Say $p$ is an odd prime, and take two matrices $A,B\in GL_n({\mathbb Z}_p)$ of finite order $m$. Is it true that they are conjugate in $GL_n({\mathbb Q}_p)$ if and only if their reductions mod $p$ are conjugate in $GL_n({\mathbb F}_p)$?

This is a follow-up to the question whether $GL_n({\mathbb F}_p)$-conjugacy implies $GL_n({\mathbb Z}_p)$-conjugacy, for which the answer is "No".

For $p=2$ this is not true, although I would be very much interested to know whether conjugacy mod 4 implies conjugacy over ${\mathbb Q}_2$.

(Note that if the answer is "Yes", then by character theory any two representations $\rho, \rho': G\to GL_n({\mathbb Z}_p)$ that are equivalent mod $p$ are equivalent in $GL_n({\mathbb Q}_p)$.)