  In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for
the sum of squares of the degrees of the irreducible $p$-Brauer characters of a
finite group $G$. We prove this conjecture for the prime $p=2$. For this we
rely on the recent reduction of Willems' conjecture to a question on
quasi-simple groups by Tong-Viet. We also verify the conditions of Tong-Viet
for certain families of finite quasi-simple groups and odd primes. On the way
we obtain lower bounds for the number of regular semisimple conjugacy classes
in finite groups of Lie type.
