Clearly, this situation is not stable. In fact, a superadiabatic gradient causes convection. Convection tends to lower the temperature gradient because the rising parcel of gas will eventually be dispersed, dumping its excess thermal and kinetic energy into its surroundings and heating up said surroundings. In stars, the convection process is known to be highly efficient, with a typical d ln ⁡ ⁡ T d ln ⁡ ⁡ P that only exceeds the adiabatic gradient by 1 part in 10 million.