The zero-location and asymptotic conjecture for the pantograph equation

For $\alpha>1$, let $f_\alpha$ be the solution of

$$y'(x)=y\left(\frac{x}{\alpha}\right),\qquad y(0)=1.$$

Let $\mathbb{A}$ be the set of zeros of $f_\alpha$.

Zero-location and asymptotic conjecture. The set $\mathbb{A}$ is countably infinite and contained in $\mathbb{R}$. Its elements can be enumerated as ${x_n}_{n\geq1}$ with $x_n>x_{n+1}$, where

$$x_n=-(n+\theta_n)\alpha^{n-1},$$

with $\theta_n\in(0,M_\alpha)$, where $M_\alpha$ is a positive constant depending only on $\alpha$ and independent of $n$. Moreover,

$$\lim_{n\to\infty}ntheta_n=\frac{\alpha}{3}\left(\ln h(\alpha)\right)',$$

where

$$h(\alpha)=\sum_{k=1}^{\infty}(2k-1)(-1)^{k-1}\alpha^{(k-k^2)/2}.$$

The conjecture concerns the complete reality of the zeros and their precise asymptotic locations for the parameterized pantograph equation; the preceding discussion establishes analogous zero-reality results in a special case, while the general assertion remains unproved in the source.