Double-sided printing works for Mac OS but not for iOS

Brother Printer MFC-J4420DW:

Brother claims that there's no printer driver available, and to use AirPrint. My MacBook Pro had no problem with double-sided printing using Monterey 12.1, but my iPhone and iPad under iOS 15.2 only prints single-sided when double-sided printing is selected. Is this Apple or Brother's lack? I think it's Brother, but wherever I go on their website, the "Contact Us" choice leads to a page full of downloads, including a firmware update link that never succeeds despite their declaration that a firmware update is available.

But still, could it be that iOS has not been updated to the point of making AirPrint work?