The LCR insures no double-spend exist. There is no fix for the idea of taking the conjunction of chains. The idea is flawed that is why I abandoned it. Also note I had added to my prior post.
That's why you'd need a new chain selection rule... what happens if you were to use to greatest set of all new transactions as the rule?
The greatest set of all new transactions would have the largest cumulative POW weight (in a system with a constant POW cost per transaction), such that any double spend by the attacker within this set would become the canonical spend, not the double spend.