Study poetry extremely lovingly? On the other hand … (10)
I believe the answer is:
'on the other hand' is the definition.
'study poetry extremely lovingly?' is the wordplay.
'study' becomes 'con' (I've seen this before).
'poetry' becomes 'verse' ('verse' can be a synonym of 'poetry'**).
'extremely' says to hollow out the word (remove centre letters).
'lovingly' with its middle removed is 'ly'.
(Another definition for conversely that I've seen is " On the other hand".)