A question related to the Isbell problem

Given two compatible metrics on a metrizable space X. It is well known that they give rise to the same Hausdorff hypertopologies and upper Hausdorff hypertopologies, on the collection of all closed subsets of X, if and only if they are uniformly equivalent. This is no longer true for the lower Hausdorff hypertopology; indeed a weaker condition is needed, and this condition has been found by Costantini and Vitolo. m of this paper is to generalize the result proved by Costantini and Vitolo to the uniform case. l also show how this result provides a new solution to the Isbell problem.