je pense qu'elles auraient éclipsé les chanteurs.

AH,  

Compound tenses conjugated with avoir don’t take past participle agreement (specifically) with the subject - which is ‘ elles ‘ in the 2nd clause.

They also do not take past participle agreement if the verb’s direct object does not precede the verb.

‘Les chanteurs ‘ is the direct object of the verb form ‘ auraient éclipsé ‘.

 ‘ Que ‘ is a conjunction in this sentence, not a direct object relative pronoun of the following verb. 

The bolded sentence as written initially is correct.

 Special cases when the past participle agrees (in number & gender) when used with 'avoir' in the compound past in French (Le Passé Composé) 

Conjugate regular -er verbs (+ avoir) in the compound past in French (Le Passé Composé)

 https://www.wordreference.com/fren/que

 https://www.lawlessfrench.com/grammar/subordinating-conjunctions/