  Recently, a new conjecture on the degrees of the irreducible Brauer
characters of a finite group was presented by the second author. In this paper
we propose a 'local' version of this conjecture for blocks B of finite groups,
giving a lower bound for the maximal degree of an irreducible Brauer character
belonging to B in terms of the dimension of B and well-known invariants like
the defect and the number of irreducible Brauer characters. We also propose a
weaker version of this conjecture for which a slight reformulation leads to
interesting open questions about traces of Cartan matrices of blocks.
  We then show that the strong conjecture is true for blocks with one simple
module, blocks of p-solvable groups and blocks with cyclic defect groups. It
also holds for many further examples of blocks of sporadic groups, symmetric
groups or groups of Lie type. We also show that the weak conjecture is true for
blocks of tame representation type.
