Erdős Problem #1132 — Pointwise growth of Lebesgue functions

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 $x_1,x_2,\ldots\in [-1,1]$ be an infinite sequence, and let $L_n(x) = \sum_{1\leq k\leq n}\lvert l_k(x)\rvert,$ where each $l_k(x)$ is defined above with respect to $x_1,\ldots,x_n$. Must there exist $x\in (-1,1)$ such that $L_n(x) >\frac{2}{\pi}\log n-O(1)$ for infinitely many $n$? Is it true that $\limsup_{n\to \infty}\frac{L_n(x)}{\log n}\geq \frac{2}{\pi}$ for almost all $x\in (-1,1)$?