Nonconstancy conjecture for the stable-domain-of-attraction coefficient

Let $\Lambda>1$ and let $\alpha\in(0,2)$. For each $\theta>0$, let $c_\theta$ denote the coefficient defined in Proposition corresponding to the asymptotics of the characteristic function of $Z$. Nonconstancy conjecture. For every $\Lambda>1$ and $\alpha\in(0,2)$, there exist $\theta_1,\theta_2>0$ such that

$$c_{\theta_1}\ne c_{\theta_2}.$$

This would show that $\theta\mapsto c_\theta$ is not constant and, by the cited proposition, would establish that $Z$ is not in the domain of attraction of a stable random variable. The statement is presented as something the authors believe, and no resolution is supplied here.