A LOO protects you from the main risk of market-on-open, which is an auction that clears far from the prior close. If the calculated opening price is worse than your limit, your order simply does not participate.
Partial participation is normal: the auction may fill part of your size at the clearing price and cancel the rest, depending on the venue's rules.
Example: prior close 50.00. You enter a LOO buy for 1,000 at 50.50. If the cross prints 50.35, you fill up to 1,000 at 50.35 — auction fills are at the single clearing price, so a limit better than the cross still pays the cross. If the cross prints 53.10, nothing fills and you keep your cash.
Related: market-on-open, auction-only-order