If more than one coin did the "mini-blockchain" proposal, would that make the weaker coin vulnerable?
While I didn't see it addressed explicitly in the paper, the attack could be defeated by requiring a competing blockchain to have common ancestry in the proof-chain in order for it to be considered. 

Out of the proposed triplet account-tree, mini-blockchain, and proof-chain, the proof-chain has by far the least memory requirement with only 512 bit (64 byte) per element. If applied to a coin with a blocktime of 1 minute, one year worth of proof-chain would only have 32 megabytes (containing 525600 proofhashes), while one year worth of proof-chain should be sufficient to allow for any valid competing chains.

The above assumes of course that each of the triplet does what it should be doing. I had one follow-up question though, posted here: Mini-Blockchain proposal: How is the proof-chain secured against injections?