Materials to reduce sidewall flex & vibration ?

Soundsrealaudio, I agree that a composite material is ideal.
I'm searching for the 'ideal' practical solid,high mass deadening
sheet and the ideal adhesive. I want to create maximum stiffness
with highest possible damping. It seems carbon filled lead epoxy
is a contender. I'm hoping someone with industrial experience may
have something to contribute. Shipbuilding, aeronautical, train building
or massive motor vibration reducing techniques perhaps. 

You should note that low mass materials resonate at higher frequencies. Ping a wine glass and you can see how that works. Now take a hammer to a heavy piece of metal. Much lower frequency. High frequencies are easier to dampen. To stop the wine glass from resonating simply touch it with your finger. This illustrates another principle important to controlling resonance and that is composite technology. Your finger resonates at a different frequency then the glass and together they work to absorb the resonance. 
You might reset your goal to making the speakers sound better by reducing the cabinet resonance. There are not a lot of speaker companies out there that do more then stuff foam in the box. A few but not many.

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.