Fake character degrees equal character degrees if algebra has nilpotency index p or less

Statement
Suppose $$F$$ is a finite field of size $$q$$, a prime power with underlying prime $$p$$. Suppose $$N$$ is a nilpotent associative finite-dimensional algebra over $$F$$ such that $$N^p = 0$$. Suppose, further, that $$G$$ is the algebra group corresponding to $$N$$. Then, the fake character degrees of $$G$$ coincide with its usual character degrees.

Related facts

 * Fake character degrees need not equal character degrees