The Canon Digital Learning Center has published an article on the just-released Canon EOS-1D X firmware version 1.1.1.
As I expected would be Canon's position on the question, the Canon EF 70-300mm f/4-5.6L IS USM Lens is officially not compatible with the Canon EF 1.4x III Extender.
As discussed here, the 70-300 L was initially included in the 1D X's f/8 AF-capable lens and extender combination list on the new Canon 1D X special firmware page. This lens is no longer included on that page - and is not included on the CDLC firmware in-depth page's compatibility list.