Y&R .. why doesn't Lauren want Phyllis to fight for custody of Lucy?

I thought she was just concerned that filing for custody would bring the fact that Lucy is actually Daisy's baby to light in Genoa City. But if that were true.. why did she immediately run and tell Gloria and Chloe.. who then filed a custody suit of her own. Lauren didn't have a problem with Chloe doing that either. Won't that bring just as much attention?