The reverse order law \((ab)^\#=b^\dagger (a^\dagger abb^\dagger)^\dagger a^\dagger\) in rings with involution (Q496358)

For the reverse order law mentioned in the title and every of the laws \((ab)^\#=b^*(a^*abb^*)^\dagger a^*\), \((a^\dagger ab)^\#=b^\dagger(a^\dagger abb^\dagger)^\dagger\), \((abb^\dagger)^\#=(a^\dagger abb^\dagger)^\dagger a^\dagger\), \((a^*ab)^\#=b^*(a^*abb^*)^\dagger\) and \((abb^*)^\dagger = (a^*abb^*)^\dagger a^*\), the authors have found a group of equivalent conditions. Additionally, some sufficient conditions for the first two laws are given. Further, presented is a group of mutually equivalent (under a certain assumption) conditions which, in conjunction with the first of these laws, are necessary and sufficient for the identity \((ab)^\#=b^\dagger a^\dagger\) to hold.