Zero-free conjecture for the auxiliary function in fractional Schrödinger equations

Let $calpha\in(1,2)$, and let $calpsi(t)$ be the function defined in the paper by equation (psidf). Zero-free conjecture. The function $calpsi(t)$ has no zeros on $(0,\infty)$, namely

$$\psi(t)\neq 0$$

for all $t>0$. If true, this would imply well-posedness of the corresponding backward fractional Schrödinger problem for all $T>0$ when $\nu=\alpha$ and $1<\alpha<2$. The conjecture remains open; the paper contrasts it with the fractional diffusion-wave case, where zeros of the corresponding function are known to exist.