Should it be "persistent [high] levels of unemployment" or "persistently high levels of unemployment"?
Solution 1:
Persistent can be employed adjectivally or adverbially.
Used as an adjective, qualifying a noun it would not take the ...ly suffix. She is suffering from a persistent cough.
Where it qualifies a verb, or an adjective, its adverbial form with the ...ly suffix is employed. He persistently argued the point or The restaurant provided persistently good food. Or, our example, The country endured persistently high levels of unemployment.
But where we speak of persistent levels of high unemployment - persistent no longer qualifies high, but the nounal phrase levels of high unemployment. So it takes its adjectival form, without the ...ly suffix.
Solution 2:
When one sees high levels, there is an automatic presumption of alternative bins that include low levels and medium levels, which together form a set. Bin labels become constituent nouns, and can take adjectives as modifiers.
In the case of persistent and persistently, they don't perfectly map the same semantics. Persistent has a considerably broader range of senses than does persistently, and here the adverb feels awkward. The meaning of stubbornly resisting attempts to remediate isn't really covered by the adverb, but is a common use of the adjective.
So I parse the phrase as [persistent [high levels]][of unemployment].
Note that persistent levels of high unemployment isn't quite the same in that it doesn't create the set of high/medium/low levels the way persistent high levels does.
Don't ignore any persistent "funny noises" as they can be the precursor of trouble. (jaguarforums.com)[https://www.jaguarforums.com/forum/xj-xj6-xj8-xjr-x350-x358-28/recommended-maintenance-schedule-04-xjr-42567/)
Residents of the city complained of a persistent “funny smell” after the incident. https://ecology.iww.org/node/965
Gender, Pulmonary Morbidity and Persistent High Concentrations of Weak Fetal Androgens in Preterm Infants (Nature.com)[https://www.nature.com/articles/pr2005423]