Let's assume a brown dwarf is on orbit around a main sequence star. Than the star becomes a red giant.
Let's assume the brown dwarf has "the right" orbit and can syphon matter from the red giant.
Is it possible for the brown dwarf to accrete enough mass to become a star itself?
By keyword search I could not find anything relevant. So I would not be surprised if the answer is "no". Maybe brown dwarfs have not enough gravity to accrete matter, and the evaporation by the red giant's radiation will prevail.
But still, I could miss some results.
Have anybody modeled such star systems?
Have anybody observed candidates for such star systems?