HUAAAANG

If Auxiliary is not Sigzil's original spren, then I would presume that it was his spren that somehow became the dawnshard. The statement "I phrased that very intentionally" seems to be an indication of a deliberately literal statement.

Upon consideration, it seems this shard may be another opportunity for some word play, much like Odium claiming Passion. If Virtuousity also represents "Virtue," either loosely or by design, it slots rather perfectly into the setting we've been shown thus far.