The execution pin/wire is, by definition, tied to the execution of the node. If the node executes correctly, the wire will pass on the execution; in other words, something isn’t executing correctly. Check for compilation errors and/or message log errors.
It’s really hard to be of any help if we don’t have the context, I’m not even sure which function(s) you’re referring to in your description of the situation (is it the node in the image, or the parent function, or something else). Please screenshot both the graph inside the Dialogue node and the parent graph (the graph your first screenshot was in).