Classify all blocks of the centraliser algebra of symmetric groups

Classify the p-blocks of the centraliser algebra kS_m^{S_l} by proving that two Specht modules S^{\mu_1\setminus\lambda_1} and S^{\mu_2\setminus\lambda_2}, for skew-partitions of sizes l and m, lie in the same p-block if and only if core_p(\mu_1)=core_p(\mu_2) and core_p(\lambda_1)=core_p(\lambda_2).

Background

The centraliser algebra kS_m^{S_l} is the principal object of the paper, and its modular block structure is described by a conjecture attributed to Ellers and Murray. The conjecture predicts that the block of a skew Specht module is determined independently by the p-cores of its outer and inner partitions. The paper proves this prediction for generalised ribbon blocks and belt blocks, but leaves the classification for arbitrary blocks unresolved.