My (very poor) layman's understanding of physics is that as something approaches c, it has a relativistic mass that is a combination of its actual mass and velocity.

Do this affect its acceleration due to gravity?

In other words, does a given "light" object and a given "heavy" object hit a relativistic speed of e.g. (.9)c at the same time if their acceleration is entirely due to gravity?

My gut says yes, but I couldn't say why. I'm only skeptical because of the other layman's physics expression along the lines of "It takes more and more energy to accelerate something as you approach the speed of light", but I suppose that energy would come from gravitation potential energy (which is already a product of mass) in the above thought experiment.