GCTA - Unbiased Estimate of Inbreeding Coefficient

Question

[kramonikov]

Hello, so I am working with GCTA (https://cnsgenomics.com/software/gcta/#Overview). In the 2010 Nature paper (https://www.ncbi.nlm.nih.gov/pubmed/20562875, online methods section) and the 2011 paper (https://www.ncbi.nlm.nih.gov/pubmed/21167468, "Estimating of inbreeding coefficients section"), they provide multiple estimates of F (the inbreeding coefficient) and state that these are unbiased. They don't provide any proof of this and I'm having a little trouble understanding how these are unbiased. Relevant section from the 2011 paper reproduced below:

How is F^III in this case an unbiased estimator of F?