Logarithmic-phase asymptotic conjecture for the orthorecursive coefficients

Let $(c_n)$ be the coefficient sequence in the orthorecursive expansion of unity. There are real constants $\delta$, $\varphi$, $A$, and $P$, and a constant $\varepsilon>0$.

Logarithmic-phase asymptotic conjecture. For all $n>0$,

$$c_n=\frac{A}{n^\delta}\sin(P\ln n+\varphi)+O(n^{-\delta-\varepsilon}).$$

This gives a precise oscillatory model in which the phase is linear in $\ln n$, explaining the suggested asymptotically geometric spacing of sign changes.