Passant Rabie (the author of this summary for Space.com) seems to think the original article is about how stars shrink down to white dwarfs (a process where they LOSE mass!)
He mentioned on at least four occasions that it GAINS mass. How did you miss that?
"As white dwarf stars gain mass, they shrink in size "
"But these stellar remnants hold a mystery, as when white dwarfs increase in mass, they shrink in size. White dwarfs therefore will end up with a mass similar to that of the sun, but packed into a body the size of the Earth. "
"The reason why white dwarfs increase in mass while shrinking at the same time ..."
'"Because the star gets smaller as it gets more massive, the gravitational redshift effect also grows with mass," Zakamska said. '
A continuous mass flow onto a WD will shrink it as it gains mass. Given enough mass increase, a WD will become a neutron star. Further mass would cause it to become a BH. [This would require the WD from becoming a Type 1a SN, admittedly.]
Just from reading the article, it seems that the reduction in size along with the increase in mass will increase the surface gravity, thus the gravitational redshift will increase. If this redshift can be tickled out of the redshift caused by peculiar motion, then it will be the redshift associated with GR.