  It has been established that Strong Cosmic Censorship Conjecture (SCCC) is
respected by an asymptotically flat black hole with charge, but violated by a
charged asymptotically de Sitter black hole. We studied the instability of the
Cauchy horizon for an accelerating black hole in Einstein theory conformally
coupled with a scalar field. The black hole is uncharged while having inner,
outer, and acceleration horizons. In the limit of vanishing acceleration,
denoted by $A \to 0$, it becomes asymptotically flat and extremal, with its
acceleration horizon vanishing as well. By exploring the perturbation of the
massless scalar field upon the accelerating black hole, whose decay rate is
governed by the quasinormal mode spectra, we show that the SCCC is violated in
the near-extremal regime and also in the $A\to 0^+$ limit. Our result is the
first observation of a black hole violating the SCCC in an almost
asymptotically Minkowskian flat regime, as well as the first example of a black
hole violating the SCCC with neither charge nor rotation.
