Monotonicity of second price auctions with bidder-specific reserve prices
Some thoughts. If this was a SPA without a reserve price, the monotonicity of the allocation rule is trivial. When the reserve price is uniform across bidders then the allocation rule is clearly weakly monotonic now; since for low bids, increasing them might not change the probability of allocation. The only thing that makes this problem different is that the reserve prices are bidder-specific. So one possibility is that you only need to show that the logic for the previous monotonicity is still valid in this setup. Probably while trying to show that the logic still goes through you will find out if there is some extra argument you will have to make in order to still have the monotonicity. 

For example, I might worry that by increasing my bid I might increase my own reserve price, or decrease the reserve price of some other bidder and thus reduce my probability of being allocated the good.