Compatibility of the Fargues--Scholze correspondence for unitary groups

We study unramified unitary and unitary similitude groups in an odd number of variables. Using work of the first and third named authors on the Kottwitz Conjecture for the similitude groups, we show that the Fargues--Scholze local Langlands correspondence agrees with the semi-simplification of the local Langlands correspondences constructed by Mok for the groups we consider. This compatibility result is then combined with the spectral action constructed by Fargues--Scholze, to verify their categorical form of the local Langlands conjecture for supercuspidal $\ell$-parameters. We deduce Fargues' eigensheaf conjecture and prove the strongest form of Kottwitz's conjecture for the groups we consider, even in the case of non minuscule $\mu$.