I’m sorry, but no historical event known as “The Treaty of Hōryūji” between the Ryukyu Kingdom and the Japanese shogunate was signed on October 20, 1548.
The Ryukyu Kingdom, known for its prosperous maritime trade and cultural exchanges, did have historical interactions with Japan, particularly with domains in Kyushu, and later with the Satsuma Domain after its invasion in 1609. However, any significant treaties or arrangements with the Japanese shogunate would be more accurately contextualized in broader regional developments across the centuries, rather than on the specified date.
For historically accurate events on the given date or more information about Ryukyu-Japan relations, more specific sources or a different query might be needed.