Number of conjugacy classes in unitriangular matrix group of fixed degree over a finite field is polynomial function of field size

Statement
Suppose $$n$$ is a natural number. Then, there exists a polynomial function $$f$$ of degree $$n - 1$$ such that, for any prime power $$q$$, the number of conjugacy classes in the  unitriangular matrix group $$UT(n,q)$$ (i.e., the unitriangular matrix group of degree $$n$$ over the finite field of size $$q$$) is $$f(q)$$.

General observations
Below, we list some general observations about the polynomial in $$q$$ giving number of conjugacy classes in $$UT(n,q)$$.

Particular cases
There is probably some general formula for these polynomials for all degrees without having to do the entire analysis of element structure and conjugacy classes, but it's unclear what this formula is.

Stronger facts

 * Degrees of irreducible representations of unitriangular matrix group of fixed degree over a finite field are all powers of field size and number of occurrences of each is polynomial function of field size
 * Conjugacy class sizes of unitriangular matrix group of fixed degree over a finite field are all powers of field size and number of occurrences of each is polynomial function of field size

Other facts about number of conjugacy classes

 * Number of conjugacy classes in general linear group of fixed degree over a finite field is polynomial function of field size
 * Number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size
 * Number of conjugacy classes in projective general linear group of fixed degree over a finite field is PORC function of field size
 * Number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size