The infinite staircase conjecture for ellipsoid embeddings into polydisks

Let $n\geq 2$ and $k\geq 0$. For each $i\geq 0$, let $A_{i,n}^{(k)}$ and $\hat{A}_{i,n}^{(k)}$ be the associated classes, and let $C_{L_{n,k}}(\alpha)$, $L_{n,k}$, $c_{i,n,k}$, $d_{i,n,k}$, and $S_{n,k}$ denote the quantities defined above. Infinite staircase conjecture. For all $\alpha\in [\frac{c_{0,n,k}}{d_{0,n,k}},S_{n,k}]$,

$$C_{L_{n,k}}(\alpha)=\sup\left\{\Gamma_{\alpha,L_{n,k}}(A)\left|A\in\bigcup_{i=0}^{\infty}\\{A_{i,n}^{(k)},\hat{A}_{i,n}^{(k)}\\}\right.\right\}.$$

This conjecture asserts that the displayed families of classes provide all the lower bounds needed to describe the infinite staircase on the stated interval.