Before revision:  In STOC'18, Assadi et al. presented a distributed algorithm for maintaining an MIS with O(1) ( amortized) round complexity and O(m^{3/4\left(\log^2 n}\right) \left(\right)  ]  Assadi et al. is deterministic; to the best of our knowledge, the state-of-the-art distributed algorithm for this problem with a message complexity of o  (m^{3/4}) incurs a round complexity of \Omega(m^{1/3 (m^{2/3) and an amortized round complexityof } under fully dynamic edge updates in the CONGEST model, where (\cdot) suppresses polylogarithmic factors. 
After revision:  In STOC'18, Assadi et al. presented a distributed algorithm for maintaining an MIS with O(1) ( amortized) round complexity and O(m^{3/4\left(\log^2 n}\right) \left(\right) ] Assadi et al. STOC'18 (m^{3/4}) incurs a round complexity of \Omega(m^{1/3 (m^{2/3) and an amortized round complexityof } under fully dynamic edge updates in the CONGEST model, where (\cdot) suppresses polylogarithmic factors.