On colocal connectedness of $F_1(X)$ in $F_2(X)$

We show that the arc is the unique chainable continuum $X$ such that $F_1(X)$ is a set of colocal connectedness of $F_2(X)$. Also, we give a non-locally-connected continuum $X$ whose hyperspace $F_1(X)$ is a set of colocal connectedness of $F_2(X)$, answering negatively a question posed by Martínez-de-la-Vega and J. M. Martínez-Montejano (2020).