Let $p$ be any prime. Let $m,n$ be positive integers with $\frac{3}{2}m < n < 2m$. Let $x$ be a positive integer with $1 \leq x < p^n$ and $\gcd(x,p)=1$.
Let $A = \{ a \in \mathbb{Z} : 1 \leq a < p^m, \, \gcd(a,p)=1 \}$ and $B = \{ b \in \mathbb{Z} : \frac{1}{2}p^m < b < p^m\}$.
Let $N$ be the number of $(a,b) \in A \times B$ that \begin{align}\tag{1} ax \equiv b \pmod{p^n}. \end{align}
Question. Can it be shown that $N \leq C p^{2m-n}$ for all $m \geq m_0$?
Here $C$ and $m_0$ are constants that may depend on $p$, but not on anything else.
I'd be interested in anything better than the trivial bound $N \leq p^m$.
For reference the trivial bound $N \leq p^m$ is established as follows. For fixed $a \in A$, those $b \in \mathbb{Z}$ that satisfy (1) have the form $b=ax+kp^n$, $k \in \mathbb{Z}$. Since $m