Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$?

On the one hand, I know the analogous fact is false for matrices over $\mathbb{Z}$ with counterexamples constructed via the Latimer-MacDuffee theorem (but the only counterexamples I've seen break down in $\mathbb{Z}_p$ for all $p$.)

On the other hand I know there are various conditions under which $GL_n(\mathbb{Q}_p)$-conjugacy implies $GL_n(\mathbb{Z}_p)$-conjugacy, as well as lifting criteria from $GL_n(\mathbb{F}_p)$, so it seems possible that it could hold here.