The answer is yes, and the proof can be found in V. J. W. Guo, J.-C. Liu, Proof of some conjectures of Z.-W. Sun on the divisibility of certain double sums, Int. J. Number Theory 12 (2016), 615-623. An arXiv version is also [available][1].


  [1]: https://arxiv.org/abs/1412.5415