The "harness" in the line appears to refer specifically to the knights and not their horses, but even what that means is obviously very open to interpretation.
I'm not sure of the historical usage of the term in English, but etymologically related terms "Harnisch" in German and "harnas" in Dutch typically refer to full suits of 15/16th century armour; and from that word I certainly would never picture something that's mainly just mail.
Of course, it's also next-to impossible to picture something independent of the usual depictions of the Swan Knights at this stage, so in my mind they will almost automatically ride some kind of barded horses (already not very fitting for Dark Age folks), and wear some mash-up of late Byzantine and 13th century western European armour. (So like the pictured cataphract, with mail covering most of the body, a tabard and a heater shield. And swan decorations, of course. A lot of swan decorations.)
While I've never pictured the Swan Knights as cataphracts, there also isn't much saying they couldn't be! Certainly fits into the Byzantine connections, and as they've existed for centuries, also doesn't actually feel "too modern" for the setting. Maybe with the horse's armour only fully on the front, tapering off towards the back to look like wings... And in the brighest of shining metal of course!