Logarithmic representation-growth conjecture for $SL_n(\mathbb{Z}_p)$

Let $C>0$ be a constant. For each $n$ and prime $p$, consider the irreducible smooth representations of $SL_n(\mathbb{Z}_p)$, with dimension measured as the vector-space dimension of the representation. Logarithmic representation-growth conjecture. For every constant $C>0$ there exists a bound $b(C)$ such that, for all $n$ and $p$, there are at most $b(C)$ irreducible smooth representations of $SL_n(\mathbb{Z}_p)$ having dimension less than $C^n$. This is proposed as a low-dimensional, logarithmic analogue of the Aizenbud–Avni representation-counting bounds; those bounds control fixed $n$ as dimension grows, whereas the conjecture seeks uniform control over $n$ and $p$ below an exponential dimension threshold.