It's not true in general. For any given...

thrust
rpm
airspeed
prop diameter
blade number

...there will be one specific blade area or blade chord which will produce the best efficiency (or equivalently, the least shaft power).

The reason is that the blade CL's are inversely proportional to the blade chord. For the best efficiency you want the best blade airfoil CL/CD, which means the blade CL wants to be "just right" -- not too large or too small, like on a wing. So the blade chord needs to be "just right" as well.

But remember that if you change any of the four parameters listed above, the best blade chord will change as well. They are all inter-related. The following proportionality is approximately correct for low pitch props, where the rotational speed dominates the flight speed:

chord ~ thrust / (blade_num * rpm^2 * dia^3 * CL)

So for example, if you increase the diameter by a factor of 1.1 (10% increase) with all else fixed, the blade chord should shrink by a factor of

1 / 1.1^3 = 0.75

or a 25% reduction. Similarly, if you decrease the rpm by a factor of 0.9 (via a larger pitch, for example), the blade chords want to change by a factor of

1 / 0.9^2 = 1.23

or a 23% increase. Note that a larger pitch should be accompanied by a wider blades, and vice versa.