Monotonicity of the normalized point-evaluation constant in p

Determine whether, for each fixed integer d ≥ 1, the function p ↦ \mathscr{C}_{d,p}/(dp/2 + 1) is decreasing on [2, ∞), where \mathscr{C}_{d,p} is the minimal constant C such that ||P||_∞^p ≤ C ||P||_p^p for all complex polynomials P of degree at most d on the unit circle (with the L^p norm on the unit circle).

Background

Beyond the sharp bound conjecture for \mathscr{C}{d,p}, the authors formulate an additional conjecture asserting monotonic decay in p of the normalized constant \mathscr{C}{d,p}/(dp/2+1).

They provide supporting evidence via proven upper bounds for certain ranges (d ≤ 4 and 2 ≤ p ≤ 4, plus all d for p ≥ 6.8) and numerical experiments, but a general proof is not known.