Nonvanishing conjecture for the Benjamin–Ono critical tau function

Let $Z$ denote the zero set of the function $\tau(T,X)$ associated with the critical Benjamin–Ono profile. Nonvanishing conjecture.

$$Z\cap\mathbb{R}^2=\emptyset,$$

i.e., $\tau(T,X)\neq 0$ for all $(T,X)\in\mathbb{R}^2$.

The conjecture would ensure that the explicitly defined critical profile is globally well defined. The source reports numerical evidence for bounded values of $X$; no proof is supplied there.