`rustic-clippy` doesn't seem to work with Flycheck under Eglot

[upbqdn]

When I run rustic-cargo-clippy in my project, I get some errors:

cargo clippy --all-targets --all-features 
   Compiling zebra-rpc v1.0.0-beta.45 (/home/m/zcash/zebra/zebra-rpc)
warning: non-binding `let` on a future
   --> zebra-rpc/src/server/tests/vectors.rs:190:5
    |
190 | /     let _ = RpcServer::start(rpc_impl, conf.clone())
191 | |         .await
192 | |         .expect("server should start");
    | |_______________________________________^
    |
    = help: consider awaiting the future or dropping explicitly with `std::mem::drop`
    = help: for further information visit https://rust-lang.github.io/rust-clippy/master/index.html#let_underscore_future
    = note: `#[warn(clippy::let_underscore_future)]` on by default

...

cargo-clippy finished at Fri Apr 25 12:53:04, duration 2.33 s

However, setting

(after! flycheck (after! rustic (push 'rustic-clippy flycheck-checkers)))

in my Doom config doesn't make Flycheck produce the errors. The value of flycheck-checkers contains rustic-clippy:

(rustic-clippy ada-gnat asciidoctor asciidoc awk-gawk bazel-build-buildifier
               bazel-module-buildifier bazel-starlark-buildifier
               bazel-workspace-buildifier c/c++-clang c/c++-gcc c/c++-cppcheck
               cfengine coffee coffee-coffeelint css-csslint css-stylelint
               cuda-nvcc cwl d-dmd dockerfile-hadolint elixir-credo emacs-lisp
               emacs-lisp-checkdoc ember-template erlang-rebar3 erlang
               eruby-erubis eruby-ruumba fortran-gfortran go-gofmt go-vet
               go-build go-test go-errcheck go-unconvert go-staticcheck groovy
               haml haml-lint handlebars haskell-stack-ghc haskell-ghc
               haskell-hlint html-tidy javascript-eslint javascript-jshint
               javascript-standard json-jsonlint json-python-json json-jq
               jsonnet less less-stylelint llvm-llc lua-luacheck lua
               markdown-markdownlint-cli markdown-mdl markdown-pymarkdown nix
               nix-linter opam perl perl-perlcritic perl-perlimports php
               php-phpmd php-phpcs php-phpcs-changed processing proselint
               protobuf-protoc protobuf-prototool pug puppet-parser puppet-lint
               python-flake8 python-ruff python-pylint python-pycompile
               python-pyright python-mypy r-lintr r racket rpm-rpmlint
               rst-sphinx rst ruby-rubocop ruby-chef-cookstyle ruby-standard
               ruby-reek ruby ruby-jruby rust-cargo rust rust-clippy salt-lint
               scala scala-scalastyle scheme-chicken scss-lint sass-stylelint
               scss-stylelint sass/scss-sass-lint sass scss sh-bash
               sh-posix-dash sh-posix-bash sh-zsh sh-shellcheck slim slim-lint
               sql-sqlint statix systemd-analyze tcl-nagelfar terraform
               terraform-tflint tex-chktex tex-lacheck texinfo textlint
               typescript-tslint verilog-verilator vhdl-ghdl xml-xmlstarlet
               xml-xmllint yaml-actionlint yaml-jsyaml yaml-ruby yaml-yamllint
               emacs-lisp-package eglot-check)

[upbqdn]

I'm still trying to figure out how I could use Eglot with rustic and get a list of errors. Has anyone managed to pull that off?