Unlike Design view, Task Pane is not functional in HTML view, because user cannot view the result visually. So, when user views HTML view, the Task Pane mode will be switched to HTML pane mode, which displays empty content. When user switches back to Design view, the previous state of Task Pane will be persisted. This means, the last opened pane mode, including its values, will be persisted when user switches back to Design view.
This behavior is also implemented in Split view. HTML pane mode will be activated when user focuses on text area section and the last opened pane mode will be activated when user focuses on editor area.