Do static sources respond to massive scalar particles from the Hawking radiation as uniformly accelerated ones do in the inertial vacuum?

We examine the recently found equivalence for the response of a static scalar source interacting with a massless Klein-Gordon field when the source is (i) static in Schwarzschild spacetime, in the Unruh vacuum associated with the Hawking radiation, and (ii) uniformly accelerated in Minkowski spacetime, in the inertial vacuum, provided that the source's proper acceleration is the same in both cases. It is shown that this equivalence is broken when the massless Klein-Gordon field is replaced by a massive one.