Elles ne connaitront jamais mon secret is the given answer. Why will savoir in the future not do? Various dictionaries suggest they are equivalent for this kind of statement.
Connaître [qch] = to know something [noun or abstract thing]
Savoir [+ clause or + infinitve] = to know [that ... /how to ...]
In the case of 'knowing a secret', 'connaître' is the right choise.
Here is a link to a lesson which you might find useful:
I hope this is helpful.
Bonne journée !
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