What is your source?
B737 Classic & NG FCOM: "When GPS position data is available, radio updating will also occur". I can't find any reference it is actually ignored.
It is true the Supp only talks about "VOR" updates, which I've found normal as DME update is standard OFF. However, some aircraft come in a different configuration, the weirdest one being "if DME is OFF, VOR and LOC is OFF even when selected ON, unless blablabla". The majority of aircraft in our fleet switch back to a standard DME OFF once power down occurs.
Our safe solution: force it all OFF prior to the approach.