After neutralization of the ketones mixture, you heat it in presence of KOH or Et3N or DBU catalyst (DABCO), under nitrogen and you monitor by TLC. May be, you get something.
That way, I had have migrated exocyclic double bond to endocyclic ketone-conjugated one, in the past.
Hint 1: Nitrogen atmosphere is necessary because these compounds are easily oxidized to 4-methylphenol upon heating. Alternatively, you can add BHT up to 1% w/w to ketones mixture.
Hint 2: Avoid excessively high temperatures and/or long reaction times because aldol type, byproducts might be formed.