  Let $m>2$ and $q>0$ be integers with $m$ even or $q$ odd. We show the
supercongruence
  $$\sum_{k=0}^{p-1}(-1)^{km}\binom{p/m-q}{k}^m\equiv0\pmod{p^3}.$$ for any
prime $p>mq$. This confirms a conjecture of Sun.
