There is no connection to the Dominion Public Building, at it's not foreseen at this point.
The connection to the Sony Centre should not be shown. That tunnel only exists for L Tower residents to access their garage under Backstage, and may someday get them in the PATH system, but that could be several years away, who knows. Still, it will not be open to the public. There's a private entrance from 777 Bay into the north tower at RoCP, but that's not shown (nor should it be) and it arguably makes a better connection with that building that, the L Tower tunnel ever will.
Backstage's PATH walkway when it finally opens many years off is above grade, and runs along the south side of the building. Best to just remove it for the time being. Similarly, the purple line running through the non-existent 141 Bay is as non-existent as that building is itself. Theoretically it will be an above grade walkway and run along the south edge of the building when it does open several years in the future.