Nonconstancy conjecture for the spectral asymptotic function

Let $s$ be the continuous function appearing in the eigenvalue-counting asymptotics

$$N(\lambda)=\lambda^D\bigl(s(\ln\lambda)+o(1)\bigr),\qquad \lambda\to+\infty,$$

for a Sturm–Liouville problem with a non-constant weight whose primitive is arithmetically self-similar. Nonconstancy conjecture. The function $s$ is not constant. This conjecture predicts that arithmetic self-similarity produces genuine oscillations in the leading spectral asymptotics for every non-constant weight of this type; the supplied text does not state whether it has been resolved.