The overhauled Roller-Coaster Conjecture for well-covered graphs of fixed order

Let $\alpha\geq2$ and $n\geq4$ be integers satisfying

$$2\alpha\leq n\leq3\alpha-2.$$

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

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

This version records the proposed shortening of the index domain for well-covered graphs of order $n$; the source presents it as a conjectural overhauled formulation.