Granath's sign conjecture for truncation errors of the Landau constants expansion

Let $G_n$ be the Landau constants, let $N=n+1$, and write the asymptotic expansion

$$\pi G_n\sim \ln(16N)+\gamma+\sum_{k=1}^{\infty}\frac{\alpha_k}{N^k}$$

with truncations

$$A_l(N)=\ln(16N)+\gamma+\sum_{k=1}^{l}\frac{\alpha_k}{N^k}.$$

Granath's conjecture. It holds that

$$(-1)^{\frac{l(l+1)}{2}}\left(\pi G_n-A_l(N)\right)<0$$

for all $n=0,1,2,\ldots$ and $l=0,1,2,\ldots$. This is equivalent to the sign rule for the truncation error $\varepsilon_l(N)=\pi G_n-A_l(N)$. The conjecture gives optimal alternating one-sided bounds of every order for the Landau constants. The paper proves this conjecture, while earlier work had established only several low-order bounds and Granath had proposed the general pattern from numerical evidence.