Erdős Problem #1130 — Nodes maximizing local Lebesgue minima

For $x_1,\ldots,x_n\in [-1,1]$ let $l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},$ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$. Let $x_0=-1$ and $x_{n+1}=1$ and $\Upsilon(x_1,\ldots,x_n)=\min_{0\leq i\leq n}\max_{x\in[x_i,x_{i+1}]} \sum_k \lvert l_k(x)\rvert.$ Is it true that $\Upsilon(x_1,\ldots,x_n)\ll \log n?$ Describe which choice of $x_i$ maximise $\Upsilon(x_1,\ldots,x_n)$.