On two families of period integrals related to Catalan's constant

The construction of a mixed motive associated with Catalan’s constant $G={\sum }_{n=0}^{\mathrm{\infty }}(-1{)}^n\mathrm{/}(2n+1{)}^2$ in [arXiv:2510.20648] led to a supply of period integrals that evaluate to linear forms in 1 and $G$. Inside this supply, the boundary of the integration domain leads to a natural 3-parameter family of period integrals. We show that this family is closely related to the family of hypergeometric period integrals considered earlier by Rivoal, Zudilin and Nesterenko.