The higher special-point obstruction conjecture

Let $i>1$, let $v_i$ be the relevant rational accumulation point, and let $b_i$ be the corresponding parameter. Let $c_{H_{b_i}}$ be the capacity function, and let $(3,1;2,1^{\times5})$ be the nonperfect exceptional class whose obstruction is equivalent to the obstruction from the 8th ECH capacity.

Higher special-point obstruction conjecture. For $i>1$, $c_{H_b}$ has no descending staircase. Furthermore, there exists $\varepsilon>0$ such that, for

$$z\in[v_i,v_i+\varepsilon),$$

the capacity function $c_{H_{b_i}}(z)$ equals the obstruction from the 8th ECH capacity, equivalently from the nonperfect exceptional class $(3,1;2,1^{\times5})$.

The claim extends the established description at $b=1/5$ to all higher indexed special points. The supplied text describes it as experimental evidence rather than a proved result.