The theory is correct, you can prevent the abyssite from changing by keeping the next tier version. You'll only get a read on the existence of the highest tier version you have, but if you read the hint, you can still track the lower tiers as well. From ffxicyclopedia:
Quote:
If the Abyssite is resonating "feebly," it is a Tier I. If it resonates "softly," it is a Tier II. If it resonates "solidly," it is a Tier III. If it is resonating "strongly," it is Tier IV.