The uniform approximation conjecture for the Boţ-Nguyen coefficients

Let $calpha>2$ and let $cbeta$ be the parameter defining the coefficient arrays $(c_{n,k})$ and $(b_{n,k})$. Uniform approximation conjecture. If

$$\beta=\frac{\alpha}{2},$$

then

$$n\max_{k\in\{0,1,\ldots,n\}}|c_{n,k}-b_{n,k}|\to0$$

as $n\to\infty$. Such an estimate would imply the Lorentz condition for $(c_{n,k})$ from the corresponding condition for $(b_{n,k})$; the supplied text does not give a proof or disproof.