Kitchen sink water backs up...

[thirdktm]

When we wash the dishes in the kitchen sink, sometimes water gurgles and backs up into the shower and/or into the bathroom sink.
We have a 2017 Melbourne 24, on the Mercedes Sprinter chassis.
I wonder if this might be caused by the sink plug in the drain in the bathroom sink, somehow not allowing air to flow out of the water lines.

[RogerR]

I have a very different model but it has a Studor Vent ( or a very cheapo knockoff) under the sink that is designed to allow air into the drain line. They have been know to fail. Not sure if there is a way to fix other than replacement.
tried to post a picture but would not work, search Studor vent.

[CAG]

Is your gray tank valve open? If not how full is it?