Depth preserving property of the local Langlands correspondence for quasi-split classical groups in a large residual characteristic

For a quasi-split classical group over a p-adic field with sufficiently large residual characteristic, we prove that the maximum of depth of representations in each L-packet equals the depth of the corresponding L-parameter. Furthermore, for quasi-split unitary groups, we show that the depth is constant in each L-packet. The key is an analysis of the endoscopic character relation via harmonic analysis based on the Bruhat--Tits theory. These results are slight generalizations of a result of Ganapathy and Varma.