My experience with air algae is that it does die off when all essential plant nutrients are pressent. Many EI tanks are dosed at even higher fertilizer levels than I am using and when everything is done right little to no algae is pressent.When right conditions are given to plants algae has less chance. But I think current algae will still do well in nutrient rich water.
Flourish was once my primary fertilizer. However it is deficient in N, P, Ca, Cu, and Zn. These nutrient deficiencies will slow or stop plant growth which will leave most of the fertilizer for the algae.It is good to find the main cause. Some amano shrimp can help with some minor hair algae but also try to fix the main cause.