Can a rigid string under tension between two fixed points vibrate?

In this study of string vibration, a string is energized with a sinusoid driver in an attempt to validate the theory that a string can vibrate in sine waves. https://hal.science/hal-01532358v2

The authors assert a rigid string cannot vibrate! They claim that if tension on a string is so great that the string is rigid, then vibration is impossible because the string must elongate as it bends. They say a rigid string would transition to vibration but only when lower tension induces elasticity (that is, the string can stretch).

A string has a lower transition point for sustained vibration, which is tension at which rigidity is established, but there is no reason to believe there is a maximum tension above which harmonic motion does not occur.

However, according to Gauss's theorema egregium, a rigid string can bend without elongation.

Rigidity is the same thing as energy conservation, and it means the string has only one degree of freedom. Clearly a theory with one degree of freedom is better than two.

You may think that the modes of vibration on a monochord are sine waves, but they are not! Sine waves are embedded in a plane. Instead, the modes are minimum surfaces of revolution. These manifolds are embedded in Euclidean space, and their curvature is not sinusoidal. It catenoidal, like a hanging rope where gravity acts perpindicular to the rope.

Mathematically, rigidity is a boundary condition on the movement of a point at x such that $\Delta X=0$

Rigidity is equivalent to energy conservation. The string cannot stretch (elongate) unless energy is added. You know, like a sinusoidal driver.

This means that a point can only move perpendicular to the string. Then sine waves are not possible.

The theory of rigid string vibration was first proposed by Taylor, but he wrote in Latin and his work was overtaken By Euler who claimed the string could have any shape you can draw. But if the string shape is arbitrary, then you don't have a solution to the wave equation!

So, my question is can you explain why a rigid string cannot vibrate? Why can't rigidity make a classical mechanical model of string action? Why is having two degrees of freedom better than one?