Relative images are not displayed when switching from Source tab to Wysiwyg tab

To reproduce, open the wysiwyg editor, and insert an image attached to the current page, say image.png, using the Image menu.
Switch to source, you should see something like

[[image:image.png]]

Switching back to the wysiwyg tab will not display the image anymore but its alt text, and the src of the image will be something like

/xwiki/bin/download/Fake/Document/image.png

This is happening because on tab switch the content of the document is re-rendered in a WysiwygService function which does not have the correct context.