The special case I${}_3$ of the Kholodenko-Silagadze multiple integral considered anew

The nested Kholodenko-Silagadze quadrature $$I_n={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}d{s}_1{\int }_{-\mathrm{\infty }}^{s_1}d{s}_2{\int }_{-\mathrm{\infty }}^{s_2}d{s}_3\cdots {\int }_{-\mathrm{\infty }}^{s_{2n-3}}d{s}_{2n-2}{\int }_{-\mathrm{\infty }}^{s_{2n-2}}d{s}_{2n-1}{\int }_{-\mathrm{\infty }}^{s_{2n-1}}d{s}_{2n}\cos (s_1^2-s_2^2)\cos (s_3^2-s_4^2)\cdots \cos (s_{2n-3}^2-s_{2n-2}^2)\cos (s_{2n-1}^2-s_{2n}^2)=\frac{2}{n!}{\left(\frac{\pi }{4}\right)}^n,$$ obtained for all integers $n\ge 1$ by an elegant but indirect argument, is tackled anew from a uniform quadrature reduction viewpoint. Along the way, at its first instance of real difficulty when $n=3,$ the recondite quadrature $${\int }_0^{\mathrm{\infty }}\frac{\cos (u)}{u}du{\int }_0^u\frac{{\sin }^2(v)}{v}dv+{\int }_0^{\mathrm{\infty }}\frac{\sin (u)}{u}du{\int }_0^u\frac{\sin (v)\cos (v)}{v}dv=\frac{{\pi }^2}{12},$$ heretofore presumably unknown, receives an indirect resolution with its indicated value of ${\pi }^2\mathrm{/}12$.