  We prove the Scholze--Weinstein conjecture on the existence and uniqueness of
local models of local Shimura varieties and the test function conjecture of
Haines--Kottwitz in this setting. In order to achieve this, we establish the
specialization principle for well-behaved $p$-adic kimberlites, show that these
include the v-sheaf local models, determine their special fibers using
hyperbolic localization for the \'etale cohomology of small v-stacks and
analyze the resulting specialization morphism using convolution.
