  James's Conjecture predicts that the adjustment matrix for blocks of the
Iwahori-Hecke algebra of the symmetric group is the identity matrix when the
weight of the block is strictly less than the characteristic of the field. In
this paper, we consider the case when the characteristic of the field is
greater than or equal to 5, and prove that the adjustment matrix for the
principal block of $\mathcal{H}_{5e}$ is the identity matrix whenever $e\neq4$.
When $e=4$, we are able to calculate all but two entries of the adjustment
matrix.
