  We introduce a new form for the bosonic channel minimal output entropy
conjecture, namely that among states with equal input entropy, the thermal
states are the ones that have slightest increase in entropy when sent through a
infinitesimal thermalizing channel. We then detail a strategy to prove the
conjecture through variational techniques. This would lead to the calculation
of the classical capacity of a communication channel subject to thermal noise.
Our strategy detects input thermal ensembles as possible solutions for the
optimal encoding of the channel, lending support to the conjecture. However, it
does not seem to be able to exclude the possibility that other input ensembles
can attain the channel capacity.
