Insertion theorems for maps to linearly ordered topological spaces

Let X   be a topological space and (Y,≤)(Y,≤) a linearly ordered topological space. Following the Katětov–Tong Insertion Theorem, a pair (X,Y)(X,Y) is said to have the insertion property if for every upper semicontinuous map g:X→Yg:X→Y and every lower semicontinuous map h:X→Yh:X→Y with g(x)≤h(x)g(x)≤h(x) for all x∈Xx∈X, there exists a continuous map f:X→Yf:X→Y such that g(x)≤f(x)≤h(x)g(x)≤f(x)≤h(x) for all x∈Xx∈X. We show that if (X,Y)(X,Y) has the insertion property for every normal space X, then Y is order isomorphic to some interval in the real line. We also prove that if X is a paracompact space or a cardinal and L   is the double edged long line, then (X,L)(X,L) has the insertion property.