Materials to reduce sidewall flex & vibration ?

Ptss, I suggest you read up on vibration theory, natural frequency and resonance before you attempt anything.  A engineering resource to be more specific.  I don't think you will gain much by just sticking other materials to mdf panels without an idea of what it would do to the resonance frequencies of your panels, might make your speakers sound worse.  

Hello rotarius. I appreciate your suggestions are very relevant so likely you
have some knowledge. Do you have some more specific suggestions? Thanks.

My final weight in. 

Buy new speakers and give up the science experiment. 

Spoken like a dealer.

Then don't buy new speakers, buy used ones on Audiogon.