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We present an extended zero-field muon spin relaxation (ZF-$\mu$SR) study ofoverdoped Bi$_{2+x}$Sr$_{2-x}$CaCu$_2$O$_{8+\delta}$ (Bi2212) single crystals,intended to elucidate the origin of weak quasistatic magnetism previouslydetected by $\mu$SR in the superconducting and normal states of optimally-dopedand overdoped samples. New results on heavily-overdoped single crystals show asimilar monotonically decreasing ZF-$\mu$SR relaxation rate with increasingtemperature that persists above the pseudogap (PG) temperature $T^*$ and doesnot evolve with hole doping ($p$). Additional measurements using an ultra-lowbackground apparatus confirm that this behavior is an intrinsic property ofBi2212, which cannot be due to magnetic order associated with the PG phase.Instead we show that the temperature-dependent relaxation rate is most likelycaused by structural changes that modify the contribution of the nuclear dipolefields to the ZF-$\mu$SR signal. Our results for Bi2212 emphasize theimportance of not assuming the nuclear-dipole field contribution is independentof temperature in ZF-$\mu$SR studies of high-temperature (high-$T_c$) cupratesuperconductors, and do not support a recent $\mu$SR study ofYBa$_2$Cu$_3$O$_{6+x}$ that claims to detect magnetic order in the PG phase.

Absence of $\mu$SR evidence for magnetic order in the pseudogap phase of Bi$_{2+x}$Sr$_{2-x}$CaCu$_2$O$_{8+\delta}$