The endpoint-maximization conjecture for the Lebesgue function of modified Chebyshev points

Let $\bar{\beta}_n=2/n$ with $n\in\mathbb{N}_{>0}$, and let $\mathcal{U}^{\bar{\beta}_n,0}_{n}=\mathcal{U}_{n+1}\setminus\{-1\}$ be the associated set of interpolation points on $\Omega$. Write $\lambda(\mathcal{U}^{\bar{\beta}_n,0}_{n};x)$ for its Lebesgue function and $\Lambda(\mathcal{U}^{\bar{\beta}_n,0}_{n},\Omega)$ for the corresponding Lebesgue constant.

Endpoint-maximization conjecture. The maximum of the Lebesgue function is attained at $x=-1$, namely

$$\Lambda\big(\mathcal{U}^{\bar{\beta}_n,0}_{n},\Omega\big)=\lambda\big(\mathcal{U}^{\bar{\beta}_n,0}_{n};-1\big)=2n-1.$$

The preceding theorem establishes the value of the Lebesgue function at $x=-1$; the conjecture asserts that this endpoint value is the global maximum over $\Omega$.