There exist many infinite series identities involving harmonic \( H_{n}=\sum_{k=1}^{n}\frac{1}{k}\) and skew-harmonic numbers \( O_{n}=\sum_{k=1}^{n}\frac{1}{2k-1}\) in the literature. In the study [\textit{X. Wang} and \textit{W. Chu}, Rocky Mt. J. Math. 52, No. 5, 1849--1866 (2022; Zbl 1510.11072)], the following open problem was proposed: For \(\lambda \) being a natural number, evaluate the infinite series below in closed form \[ W\left( \lambda \right) :=\sum_{n=1}^{\infty }\binom{2n}{2}^{2}\frac{O_{n}}{ 16^{n}\left( 1+2n-2\lambda \right) ^{2}}. \] The main aim of this study is to give a full solution to this problem. An analytic solution (Theorem 2.4) was achieved by incorporating partial fraction decompositions, recurrence relations and hypergeometric series evaluations, using Catalan constant, harmonic numbers, skew harmonic numbers, and squared central binomial coefficients.