The Chebyshev-distance asymptotic conjecture for hypergraph clique-shadows

For each $k\geq 3$, let $\Delta_k$ be the maximum Chebyshev distance between the coordinatewise powers of the principal eigenvectors of a connected $k$-graph and its clique-shadow:

$$\Delta_k:=\max_{H\in\mathcal{H}(k)}D(\hat y,\hat x).$$

Here $D(\hat y,\hat x)=\max_v|y_v^k-x_v^2|$, where $y$ and $x$ are the principal eigenvectors of $H$ and $\partial^*(H)$, respectively, and $\hat y=(y_v^k)_{v\in V}$ and $\hat x=(x_v^2)_{v\in V}$. Chebyshev-distance conjecture.

$$\Delta_k=\frac12+o(1)$$

for all $k\geq 3$. The surrounding theorem proves the upper bound $D(\hat y,\hat x)\leq 1/2$ and establishes only the asymptotic lower bound $\lim_{k\to\infty}\Delta_k=1/2$; the stated uniform asymptotic formula is not resolved in the supplied text.