Fizz-Keeper

[2] The kinetics of the loss of "fizz" (CO2) after pouring out part of the liquid and resealing the bottle is somewhat complicated, but the computation of the final equilibrium concentration of CO2 in the liquid and the gas phase can be done (with or without pumping air into the bottle) using Dalton's law (a consequence of ideal gas theory).

This shows[1][2][3][4] that the equilibrium CO2 concentrations are independent of the air pressure -- the device does not prevent loss of CO2 from the liquid.

There is more evidence of this in Henry's Law, which shows that partial pressure in the container must be not only restored, but with the same gas.

Studies of the kinetics,[2][3][4] however, do report that pumping in air pressure slows down the rate at which dissolved CO2 comes out of solution.

Apparently the application of the Fizz-Keeper with pressurized air can extend this process for a short time, but not for days.