Constant local Hölder exponent conjecture for refinable functions

Let $\varphi$ be a refinable function, and suppose that its local Hölder exponent is constant and equal to $\alpha$. Constant local Hölder exponent conjecture. Then

$$\alpha=1.$$

The paper studies refinable functions whose local regularity does not vary. The supplied text does not state whether this claim has been proved or disproved.