Do engines require their extra power bars to be powered to influence FTL recharge rate?

When you upgrade your Engine Systems, it increases the rate your FTL drive charges during hostile situations. To reap this benefit, do your upgraded engine's need to be MORE powered than otherwise or is it enough to upgrade them and use only 1 bar of power?

Yes, the recharge rate of your engines will only increase to the level you are powering them. The Nesasio (Stealth) starts with level 4 engines, but there is a noticeable difference in recharge rate between 1 power and 4 power.

This is consistent with all powered systems. To get the described bonus from a certain level, you must power the system to that level.