It is important to separate the notions of validity (“does this chain satisfy all validity rules”) and being active (“among all valid chains, does it have the most work?”).
Let’s say the two branches of the chain are A (created by Bitcoin miners) and B (created by BIP110 miners). A has ~10x more work than B, but over time, after retargetting, will have a roughly equal number of blocks.
To BIP110-enforcing nodes, only B is valid, so their active chain is necessary B. They do not even “see” the A chain.
To Bitcoin nodes, both A and B are valid. However, regardless of the number of blocks in each, or difficulty adjustment, the A chain has clearly more proof-of-work than B (which is computed roughly as sum of the difficulties across all its blocks). Thus, Bitcoin nodes will never consider B to be the active chain, because A has just far more work. “being active” is in practice what matters for users, who will just not see B.
However, if the B chain somehow managed to get the most work at some later point (which inevitably means overtaking A in difficulty), Bitcoin nodes will switch over to it.
The difficulty adjustment rule (a part of the validity rules) applies per branch (as do all consensus rules): each branch’s difficulty requirement is a function just of the blocks within that same branch. The existence of another branch with another difficulty is irrelevant for validity. Of course, the “pick most-work chain” still cares about the difference in difficulty between the branches (indirectly, through the total PoW they contribute to their respective branches).
Hard forks are unrelated; those are changes to consensus rules that are incompatible in both directions. That is not the case here, the BIP110 chain is valid to Bitcoin nodes, just inferior in proof-of-work as long as its difficulty is low.
