On degrees of irreducible Brauer characters

Based on a large amount of examples, which we have checked so far, we conjecture that $\vert G\vert _{p'}\le\sum_\varphi \varphi(1)^2$ where $p$ is a prime and the sum runs through the set of irreducible Brauer characters in characteristic $p$ of the finite group $G$. We prove the conjecture simultaneously for $p$-solvable groups and groups of Lie type in the defining characteristic. In non-defining characteristics we give asymptotically an affirmative answer in many cases.