James's Conjecture holds for weight four blocks of Iwahori–Hecke algebras

James's Conjecture suggests that in certain cases, the decomposition numbers for the Iwahori–Hecke algebra of the symmetric group over a field of prime characteristic (and in particular, the decomposition numbers for the symmetric group itself) coincide with the decomposition numbers for a corresponding Iwahori–Hecke algebra defined over C, and hence can be computed using the LLT algorithm. We prove this conjecture for blocks of weight 4.