Euler solved the famous Basel problem and discovered that Riemann zeta functions at positive even integers are rational multiples of powers of $\pi$. Multiple zeta values (MSVs) are a multi-dimensional generalization of the Riemann zeta values, and MZVs which are rational multiples of powers of $\pi$ is called Eulerian MZVs. In 1996, Borwein-Bradley-Broadhurst discovered a series of conjecturally Eulerian MZVs which together with the known Eulerian family seems to exhaust all Eulerian MZVs at least numerically. A few years later, Borwein-Bradley-Broadhurst-Lisonek discovered two families of interesting conjectural relations among MZVs generalizing the previous conjecture of Eulerian MZVs, which were later extended further by Charlton in light of alternating block structure. In this talk, I would like to present my recent joint work with Minoru Hirose concerning block shuffle relations that simultaneously resolve and generalize the conjectures of Charlton.