The Roller-Coaster Conjecture for 1-well-covered graphs

Let $G$ be a $1$-well-covered graph, let $\alpha(G)$ be its independence number, let $n=|V(G)|$, and write $I(G;x)=\sum_k s_kx^k$ for its independence polynomial. Roller-Coaster Conjecture for 1-well-covered graphs. For every permutation $\sigma$ of the set $\{\left\lceil \frac{2\alpha}{3}\right\rceil,\ldots,\alpha\}$, there exists a $1$-well-covered graph $G$ with $\alpha(G)=\alpha$ and $|V(G)|=n$ such that

$$s_{\sigma(\left\lceil \frac{2\alpha}{3}\right\rceil )}<s_{\sigma(\left\lceil \frac{2\alpha}{3}\right\rceil +1)}<\cdots<s_{\sigma(\alpha)}.$$

The paper motivates this proposal by proving that the chaotic interval for $1$-well-covered graphs can be shortened to indices beginning at $\left\lceil 2\alpha/3\right\rceil$; the proposed extension itself is not resolved in the supplied text.