Fake character degrees need not equal character degrees

Statement
It is possible to have the following: $$F$$ is a finite field of size $$q$$, a prime power with underlying prime $$p$$. $$N$$ is a nilpotent associative finite-dimensional algebra over $$F$$. $$G$$ is the algebra group corresponding to $$N$$, and the fake character degrees of $$G$$ do not agree with the character degrees of $$G$$.

Related facts

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