What is the limit of the alternating series $F(z)=\sum_{n=1}^\infty(-1)^n z^{T_n}$ as $z\to1$ for a sequence $T_n\sim cn$?

Let $(T_n)_{n>0}$ be an increasing sequence of positive integers and $c>0$ a positive number such that $\lim_{n\to\infty}\frac{T_n}{n}=c$. Write $F(z)=\sum_{n=1}^\infty(-1)^n z^{T_n}$ for a real number $z\in(-1,1)$. What is the limit of $F(z)$ as $z\to1$?

For example, if $T_n=cn+d$ for some integer $d$, then it is easy to see that $F(z)\to-\frac{1}{2}$ as $z\to1$.

Is this true ($F(z)\to-\frac{1}{2}$ as $z\to1$) for any sequence $(T_n)_n$ satisfying the condition stated in the first paragraph? Do you know any counterexamples (if this is not true)? Any help or references would be appreciated!