I found that the only way to solve this problem was to make it implausible (nearly impossible) to do the bad things that control over mining can do by incorporating Byzantine fault detection and combine this with unprofitable mining, so only those who have an incentive to maintain the Nash equilibrium actually mine (so that the risk cost of attempting attacks is far too high because it isn't cost-free to try over and over as it is in the nothing-at-stake of PoS[1] and the chance of succeeding is very low due to the Byzantine fault detection).
So that in a nutshell is my design solution. The technological challenge is in the details of accomplishing it.
Thanks for these interesting hints. It seems that in your model there is an explicit or implicit penalty for not mining as there's no positive reward for mining (in that case mining would be profitable).