Four-periodic infinite staircase conjecture for four-dimensional polydisks

Let $n\in\mathbb{Z}_{\geq 2}$ and define

$$\beta_n=\frac{1}{2}+\frac{(2n+1)\sqrt{n(n^3+2n^2-1)}}{2n(n+1)}.$$

Let $c_{\beta_n}$ denote the capacity function associated with the four-dimensional polydisk, and let $\operatorname{accum}(\beta_n)$ denote the accumulation point of its infinite staircase. Four-periodic infinite staircase conjecture. The function $c_{\beta_n}$ has an infinite staircase, and it is four-periodic, with

$$\operatorname{accum}(\beta_n)=[\{2n+4,2n+2,2n,2n-2\}^{\infty}].$$

This is expected from experimental evidence and from combining the conjectured generalization in the paper with a theorem of McDuff, McDuff, and Weiler. The claim extends the specific four-periodic infinite staircase established in the paper to every integer $n\geq 2$; its general case remains open.