In version 3.9 of the API, I was able to use the isShowing method (popup-amd | API Reference | ArcGIS API for JavaScript) to see if the popup was being displayed and, if it was, to hide it as part of a map.on("click"...) event handler. Now, with 3.10, I noticed that isShowing is always true, even if the popup is not visible.
What can I do to workaround this problem?
