  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 $\mu$.
