When is the complement of the diagonal of a LOTS functionally countable?

In a 2021 paper, Vladimir Tkachuk asked whether there is a non-separable LOTS $X$ such that $X^2\setminus \{⟨x,x⟩\text{\hspace{0.17em}⁣}:x\in X\}$ is functionally countable. In this paper we prove that such a space, if it exists, must be an Aronszajn line and admits a $\le 2$-to-$1$ retraction to a subspace that is a Suslin line. After this, assuming the existence of a Suslin line, we prove that there is Suslin line that is functionally countable. Finally, we present an example of a functionally countable Suslin line $L$ such that $L^2\setminus \{⟨x,x⟩\text{\hspace{0.17em}⁣}:x\in L\}$ is not functionally countable.