The validation tool I’ve been using (the WAND browser extension by WebAIM) is based primarily on the WCAG standards, which I think WordPress is now requiring for core code. (I’m not a developer, so I’m not up on all that.)
Since WAND is an automated tool, it has its limitations, but I think the H4 tags do present a couple of potential accessibility problems that I hadn’t thought about before.
The issue is that screen readers and text-to-speech readers use the section heading levels to parse how each post’s content is organized so that the software knows how to read the content to the user in an order that makes sense. So, if some heading levels are skipped or duplicated, the reader software can get confused and might, for example, skip from the main column text to a sidebar widget and then back (especially if the main column also has H4 and H5 tags).
That can be mitigated somewhat by using HTML5 and/or ARIA attributes to distinguish the content area from the sidebar, but that wouldn’t necessarily help if people have widgets before and after posts. The ideal solution would probably be to change the widget titles from H4 to DIV. That way, they wouldn’t implicate the order of the main content text or impose any hierarchy on the widgets.
I hadn’t thought about this before — I’ve been guilty of using heading level tags as a convenient formatting shortcut rather than a hierarchical tool — but it’s something to consider for future versions.