I heard that folklore too.
I don't know how true it is, but I did read in Kalidasa's Raghuvamsam that King Janaka continued to be a confidante and advisor to Rama, and Rama gave him the position of father and respected him, so I'm sure there were still cordial relations between Mithila and Ayodhya. After all, Mithila's other daughter Urmila was still wed to Lakshman, as well as Mandavi and Shrutakirti (though one can argue they are Sankasya nagari's daughters). Mithila would not insult Ayodhya when one of its princesses is still living a happily wedded life there.
I'm sure Janak and Sunaina mourned Sita's loss in private, but Janak was also a Rajarshi. He was at that point beyond human attachments, so I guess he looked at the whole situation more philosophically.