If the homogeneous-Ω assumption is violated, then there are different types of SNPs with different Ω’s. Because MTAG combines the GWAS estimates using the genome-wide (i.e., across-SNP) variance-covariance matrix, in general the MTAG estimates will be biased in finite samples. For a type of SNP that is null for one trait but non-null for other traits, the effect estimate on the first trait will be biased away from zero. For that reason, the FDR will be inflated.
