Does the UK have any jurisdiction over the Captain Cook monument at Kealakekua Bay on the Big Island of Hawaii?

A large white stone monument was built on the north shore of the bay in 1874 on the order of Princess Likelike and was deeded to the United Kingdom in 1877. The chain around the monument is supported by four cannon from the ship HMS Fantome; they were placed with their breaches embedded in the rock in 1876. It marks the approximate location of Cook's death.
https://en.wikipedia.org/wiki/Kealakekua_Bay#Decline

The status of the land where the monument stand is not clear (it may not be UK sovereign land). The two private citizens who owned the land sold it to the UK Consul General in 1877.
https://www.jstor.org/stable/1794586?seq=1#page_scan_tab_contents

The monument was erected in 1874 and the land was bought by the British consul general in Hawaii in 1877 for $1. It has since been maintained by the Royal Navy and ships from allied fleets, as well as by a caretaker paid by the British embassy in Washington. There is some doubt over whether the monument strictly belongs to the Government or the descendants of the consul who bought it and it is believed some in the Foreign Office would welcome a developer who would guarantee its upkeep and access from the sea. A Foreign Office spokesman said: "The precise legal status of the land on which the Cook memorial stands is uncertain."

https://www.independent.co.uk/news/world/asia/captain-cooks-little-corner-of-hawaii-under-threat-from-new-golf-course-623120.html

The site was refurbished in August 2012 by the crew of HMCS Vancouver. Pictures here:
https://www.captaincooksociety.com/home/detail/captain-cook-monument-at-kealakekua-cleaned-and-repaired