About the properties of the zeros of the “perturbed ” exponential function $\sum_{n\geq 0} q^{n(n-1)/2}z^n/n!$

For $|q|\leq1$ the “perturbed” (also “deformed”/“parametric”) exponential function is a unique analytic solution of the functional equation $f'(z)=f(qz)$ satisfying the initial condition $f(0)=1$. In literature, this function appeared many times due to its connection to the exponential function. G. Valiron wrote that it is a simplest entire function after $e^x$. It turns to be related to the Tutte polynomials, its power series evidently resembles the power series of the Jacobi theta-function. Nevertheless, although all zeros of the theta-function are well known (and simple), we still cannot fully describe the properties that have the zeros of the perturbed exponential function in general case.
In 2009, A. Sokal stated a conjecture that all zeros of the perturbed exponential function are simple (as well as a number of further conjectures). This fact would be useful for studying the dynamics of the zeros as a function of the complex parameter $q$. In this talk, I would like to present the results that I know on this conjecture, as well as a few related questions.