Decoupling between gravitationally bounded systems and the cosmic expansion

Recently, it was hypothesized that some supermassive black holes (SMBHs) may couple to the cosmic expansion. The mass of these SMBHs increase as the cubic power of the cosmic scale factor, leaving the energy density of the SMBHs unchanged when the universe expands. However, following general principles of general relativity, namely, locality or junction conditions, we show that the inner part of a gravitationally bounded system is unaware of the cosmic expansion, since the outer solution can be smoothly replaced by an asymptotically flat background. For the same reason, the direct reason that we do not expand with the expansion of the universe is not that we are bound states, but rather we are positioned in a greater gravitationally bounded system, namely the local group.