Nikiel's conjecture

Let $X$ be a compact Hausdorff space.

Call a linearly ordered set $(L,<)$ with its open-interval topology a linearly ordered topological space, and say that $X$ is a continuous image of a compact linearly ordered space if there exist a compact linearly ordered topological space $L$ and a continuous surjection $f\colon L \to X$.

Call a $T_1$ space $X$ monotonically normal if there is an operator $H$ assigning to each pair $(p,U)$, where $U \subseteq X$ is open and $p \in U$, an open set $H(p,U)$ such that

1. $p \in H(p,U) \subseteq U$; 2. if $U \subseteq V$ are open and $p \in U$, then $H(p,U) \subseteq H(p,V)$; 3. if $p \neq q$ and $U, V$ are open with $p \in U$, $q \in V$, then

$$H(p,U) \cap H(q,V) \neq \emptyset \implies p \in V \text{ or } q \in U.$$

Then $X$ is a continuous image of a compact linearly ordered space if and only if $X$ is monotonically normal.