Not this old chestnut again.

Categorically, all F passengers on a Star operated flight are entitled to access SIN SKL F.

All J passengers on a Star operated flight are entitled to access SIN SKL J.

The only lounge out of bounds for non-SQ passengers is the Private Room.

They are very aware of the rules at the entrance and will not give you any problems, other than to perhaps suggest it might be best to use the lounge in the correct terminal or to remind you that they will not announce boarding for your flight.