Thought by both to be vacillating (2,3,5)
I believe the answer is:
in two minds
'both to be vacillating' is the definition.
I don't know anything about this answer so I cannot judge whether it can be defined by this definition.
'thought' is the wordplay.
I cannot quite understand how this works, but
'thought' could be 'mind' (mind is a kind of thought) and 'mind' is located in the answer.
This explanation may well be incorrect...
'by' is the link.
Can you help me to learn more?
(Other definitions for in two minds that I've seen before include "Wavering, undecided" , "where similar ideas may be foolish or clever" , "The outcome is uncertain" , "like Dr Jekyll at times?" , "Torn" .)