Cleavability and scattered sets of non-trivial fibers

In this paper, we provide a partial answer to a problem posed by A.V. Arhangelʼskiĭ; we show that if X is a compactum cleavable over a separable linearly ordered topological space (LOTS) Y such that for some continuous function f from X to Y, the set of points on which f is not injective is scattered, then X is a LOTS.