No, for this you need a minimum power speed.
The way that I picture the situation is that an aircraft has to produce a force (let's call it thrust) to oppose the drag force.
This Thrust force uses up stored energy.
To stay in the air as long as possible we need to use the energy as slowly as possible. This is done by using the minimum thrust.
In order to use the minimum thrust at a constant speed the drag must also be at a minimum at that speed.
Thus isn't your minimum power speed also the minimum drag speed.
So looking at the total drag curve (not including CL) there will be an airspeed where total drag is at a minimum. Faster or slower than this speed causes more drag.
Is it not true then to say that when flying at this minimum drag speed minimum thrust will be required since to fly faster or slower requires more thrust to offset the increased drag?
This is a basic foundation in powered flying regarding endurance, minimum sink and the old 2 speeds fro every power setting except when at min-drag.
Regards,
DFC