  The combination of strong disorder and interactions in closed quantum systems
can lead to many-body localization (MBL). However this quantum phase is not
stable when the system is coupled to a thermal environment. We investigate how
MBL is destroyed in systems that are weakly coupled to a dephasive Markovian
environment by focusing on their entanglement dynamics. We numerically study
the third R\'{e}nyi negativity $R_3$, a recently proposed entanglement proxy
based on the negativity that captures the unbounded logarithmic growth in the
closed case and that can be computed efficiently with tensor networks. We also
show that the decay of $R_3$ follows a stretched exponential law, similarly to
the imbalance, with however a smaller stretching exponent.
