Tab component may not be running filters

Steps:
1. Install the Footnotes → module that allows the user to add a filter to a text format.
2. Add the Footnotes filter to the Full HTML text format.
3. Add a custom tab, set the text format to Full HTML and add the following:

[fn]This is a footnote[/fn]
[footnotes]

However, the footnote did not show up at the bottom of the tab, as expected. It appears that the filter is not running.