  Given a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a connected
$p$-adic reductive group $G/L$, and a smooth irreducible representation $\pi$
of $G(L)$, Fargues-Scholze recently attached a semisimple Weil parameter to
such $\pi$, giving a general candidate for the local Langlands correspondence.
It is natural to ask whether this construction is compatible with known
instances of the correspondence after semisimplification. For $G =
\mathrm{GL}_{n}$ and its inner forms, Fargues-Scholze and
Hansen-Kaletha-Weinstein showed that the correspondence is compatible with the
correspondence of Harris-Taylor/Henniart. We verify a similar compatibility for
$G = \mathrm{GSp}_{4}$ and its unique non-split inner form $G =
\mathrm{GU}_{2}(D)$, where $D$ is the quaternion division algebra over $L$,
assuming that $L/\mathbb{Q}_{p}$ is unramified and $p > 2$. In this case, the
local Langlands correspondence has been constructed by Gan-Takeda and
Gan-Tantono. Analogous to the case of $\mathrm{GL}_{n}$ and its inner forms,
this compatibility is proven by describing the Weil group action on the
cohomology of a local Shimura variety associated to $\mathrm{GSp}_{4}$, using
basic uniformization of abelian type Shimura varieties due to Shen, combined
with various global results of Kret-Shin and Sorensen on Galois representations
in the cohomology of global Shimura varieties associated to inner forms of
$\mathrm{GSp}_{4}$ over a totally real field. After showing the parameters are
the same, we apply some ideas from the geometry of the Fargues-Scholze
construction explored recently by Hansen, to give a more precise description of
the cohomology of this local Shimura variety, verifying a strong form of the
Kottwitz conjecture in the process.
