(setq filename "/sshx:localhost:~")
(setq tramp-verbose 0)

;; Make sure we have a connection.
(find-file filename)
;; Check case-insensitivity once to ensure the remote check has run.
(file-name-case-insensitive-p filename)

(message "* file-name-case-insensitive-p (local)")
(benchmark 1000 `(file-name-case-insensitive-p "~"))
(message "* file-name-case-insensitive-p (remote)")
(benchmark 1000 `(file-name-case-insensitive-p ,filename))
(message "* tramp-handle-file-name-case-insensitive-p")
(benchmark 1000 `(tramp-handle-file-name-case-insensitive-p ,filename))
(message "* tramp-find-foreign-file-name-handler")
(benchmark 1000 `(tramp-find-foreign-file-name-handler ,filename))