Don't think this could be a feature provided by PGC wehen it is not integrated within geocaching.com.
See related discussion https://project-gc.com/qa/?qa=24507/why-arent-lab-caches-visible-on-geocaching-com
AL are (at the moment) different from GC. So there would be an interface needet from PGC to AL. And AL would need to provide all this information with an api for PGC.
So would reccomend to change this to feature request but even than I think it will need some more actions from GC.
Startkacher already provided the information: Geocaching President Bryan Roth just said on the Geocaching Vlogger's Youtube Gathering from July 15th that Adventure Labs will likely be on the Geocaching.com site by the end of this year. https://www.youtube.com/watch?v=BGrUMEpSfF8 (at 59:20 minutes)