No

unless you are saying that the loads going into the flexy tube are greater than the stiff tube
because the pulse of the pedal load due to rider enertia loads the flexy tube more than the stiff tubes

http://hyperphysics.phy-astr.gsu.edu/hbase/pespr.html

Elastic potential energy is Potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring. It is equal to the work done to stretch the spring

"a" Mount one end of a stiff tube to the bench
"b" Mount one end of a flexy tube to the bench

Load, stress both tubes same amount at the free end.

"a" tube deflects a certain amount {amount of strain}
"b" Tube deflects a certain amount, and yes it more than "a" tube {amount of strain}

Both have the same amount of stored energy.
Both will release this energy in same amounts if harnessed in some way. {bike frame and chain etc}
Stiff tube did not lose potential energy when under stress

How the flexy tubes increases a bike rider's biomechanical efficiency is yet to be demonstrated by those with the planing theory belief.