A brief update to this post from 2016.
Several Japanese news outlets reported today that one of the swords that were discovered during 1939 repair measures of the Kasuga-taisha turned out to be a Ko-Hōki (古伯耆) from the end of the Heian period, possibly a Yasutsuna (安綱), dating back to the birthplace of the nihontō. The blade was polished recently, is ubu I think, and has a nagasa of 82.4 cm. Its kokushitsu-tachi koshirae (黒漆太刀拵) dates to the 14th century and it is assumed that the sword was offered to the shrine some time in the Nanbokuchō or at the latest at the beginning of the Muromachi period. Below are some links with pictures and please also check out the video that the Nara Television Channel put on YouTube.