I searched WolframAalpha and I can tell you you are partially right. The universe macroscopically is flat so they remain in proximity to the black hole on the x

I searched WolframAalpha and I can tell you you are partially right. The universe macroscopically is flat so they remain in proximity to the black hole on the x,y dimensions but move arbitrarily far on the z dimension, which is how they bend spacetime and why they attract everything around them like if they have gravity. :-)