Growth conjecture for the original Treschev problem

Consider the original Treschev problem, with coefficients $q_{2n}$ and a Diophantine rotation number. Here, $k>0$ is arbitrary and $C(k)$ denotes a constant depending on $k$. Treschev growth conjecture. If the rotation number is Diophantine, then for every given $k>0$ there exists a constant $C(k)$ such that

$$|q_{2n}| \le e^{(n \log n)/k}e^{C n}.$$

The source presents this as a numerical-analysis-based conjecture because the convergence question for the original problem remains open when $q_3=0$; the claimed bound is intended to describe the growth in that degenerate case.