Van Kampen-like property for unicoherence

Question

A space $X$ is unicoherent if whenever $A,B$ are closed connected subsets of $X$ such that $A \cup B = X$, their intersection $A \cap B$ is connected. The survey "A Survey on Unicoherence and Related Properties" by Garcia-Maynez and Illanes says that intuitively, a unicoherent space is a space with no "holes". For example, the closed disk in the plane is unicoherent, but the circle $S^1$ is not. This sounds like simple connectedness. For simple connectedness, by van Kampen's theorem, we have the following: if a space $X$ has simply connected open subsets $U,V$ such that $U \cup V = X$ and $U \cap V$ is nonempty and path-connected, then $X$ is simply connected. It is then natural to ask a similar question for unicoherence. So, my question is: if a space $X$ has closed unicoherent subsets $A,B$ such that $A \cup B = X$ and $A \cap B$ is connected, is $X$ necessarily unicoherent? If necessary, you can assume that $A,B$ are open rather than closed, or replace unicoherence conditions with open unicoherence conditions. Also, you can assume that $A \cap B$ is nonempty if needed. Thank you in advance!

Answer 1

If $X$ is compact and metric (or only Hausdorff) and $∩$ is connected and locally connected, then $X$ is unicoherent. You can search for the paper 'Special Unions of Unicoherent Continua'. But if $∩$ is connected and not locally connected, then $X$ is not necessarily unicoherent.