Does the flatness of the velocity derivative blow up at a finite Reynolds number?

A tentative suggestion is made that the flatness of the velocity derivative could reach an infinite value at finite (though very largo) Reynolds number, with possible implications for the singularities of the Navier-Stokes equations. A direct test of this suggestion requires measurements at Reynolds numbers presently outside the experimental capacity, so an alternative suggestion that can be tested at accessible Reynolds numbers is also made.