A guess on two increasing rational approximations to $\frac{\pi}{4}$

Question

When investigated the Wilf function $$W(z)=\frac{\arctan\sqrt{2\operatorname{e}^{-z}-1}\,}{\sqrt{2\operatorname{e}^{-z}-1}\,},$$ see the preprint [1] below, I proposed a guess which reads that the sequences \begin{equation}\label{approx2pi-guess-qi}\tag{QG} \frac{1}{2^k}\sum_{j=0}^{k-1}\frac{(-1)^j}{2j+1} \sum_{\ell=0}^{k-j-1}\binom{k}{\ell} =-\frac{1}{\binom{2k}{k}}\sum_{\ell=1}^{k} (-1)^{\ell} \binom{2k-\ell}{k} \frac{2^{\ell/2}}{\ell} \sin\frac{3\ell\pi}{4} \end{equation} are increasing in $k\in\mathbb{N}$ and tend to $\frac{\pi}{4}$ as $k\to\infty$.

If this guess were verified to be true, then the sequences in \eqref{approx2pi-guess-qi} would be two increasing rational approximations of the irrational constant $\frac{\pi}{4}$.

This guess is also related to a closed-form formula of the Gauss hypergeometric series ${}_2F_1\bigl(k+\frac12,k+1;k+\frac32;-z^2\bigr)$ for $k\in\mathbb{N}$, which is established in a forthcoming paper authored by me.

Question: is the above guess true?

1. Feng Qi and Mark Daniel Ward, Closed-form formulas and properties of coefficients in Maclaurin’s series expansion of Wilf’s function composited by inverse tangent, square root, and exponential functions, arXiv (2022), available online at https://arxiv.org/abs/2110.08576v2.