I don't have any personal experience on this route. But all the domestic routes I have been booking recently have had their lowest offer price points when Z bucket was open, and would go up and down in price largely in concert with changes in J/C/D/I/Z bucket availability on flights (which I check regularly on EF when there is a price change in upgrade pricing). These are the DL fare classes for both D1 and FC fares. Usually, when a flight gets down to only having inventory in the J bucket, DL has taken away the online seat upgrades (seats are X'ed out) even though there are still seats available for sale (albeit only at very high fares due to only have J bucket open).

If the flights are far out enough, you can generally see what buckets are open by simply pricing out a new ticket on flight and seeing what fare class is available for sale. Closer in, you will generally need an EF account to see what is open due to advance purchase requirements for cheaper fares like I and Z. LAX-HND is a bit unusual because the cheapest fare filings on route are I fares and there are currently no Z fares (so the only way to see Z inventory currently is with an EF account).