At minimal power speed, the consumed power (and thus, to some extent, the fuel consumption) increases with decreasing density.
Thus climbing brings a double penalty.
The only benefit of altitude is the increase in engine efficiency,
... for power needs that also increase.
The savings brought by altitude is over a distance, not over time.
Vmp = [ (4/3) . (W/S)² . (1/rho²) . (1/CD) . (1/(Pi.e.AR)) ]^1/4
Or Vmp² = k . (W/rho)
And Pmp = Dmp . Vmp ~= [(½ . CD .
rho . Vmp² . S) + (1/(Pi.e.AR) . W² . 1/(½ .
rho . Vmp² . S)] . Vmp
Obviously, Dmp, drag at minimal power speed, is constant through density variations as Vmp² is inversely proportional to density.
Pmp = k’ . (W/rho)
Pmp increases with decreasing densities
(with Vmp=minimal power speed, W=weight, S=wing area, rho=density, CD=parasitic drag coefficient, 1/(Pi.e.AR)=induced drag coefficient, Dmp=drag at minimal power speed, Pmp=power at minimal power speed, k and k' indicate a constant value)
Note: What I wrote above is only valid for airplanes whose endurance is linked to power consumption (ie: piston and turboprop planes).
For jet airplanes, the endurance is related to thrust generation with max endurance matching the minimal drag condition.
As the thrust-specific fuel consumption (TFSC) is slightly decreasing with altitude, there is a small benefit in holding at a higher altitude if you don't have to consume more fuel to climb there. I tried the NASA jet engine simulator at
https://www1.grc.nasa.gov/beginners-...cs/enginesimu/ and found that the change in TFSC, when keeping a constant IAS, more or less matches the thermodynamic efficiency variations derived from the inlet temperature and the turbine exit temperature.