  In [PRL, 113, 050401 (2014)] the authors have shown that instead of
L$\ddot{u}$der rule, if degeneracy breaking von Neumann projection rule is
adopted for state reduction, the quantum value of three-time Leggett-Garg
inequality can exceed it's L$\ddot{u}$ders bound. Such violation of
L$\ddot{u}$ders bound may even approach algebraic maximum of the inequality in
the asymptotic limit of system size. They also claim that for
Clauser-Horne-Shimony-Holt (CHSH) inequality such violation of L$\ddot{u}$ders
bound (known as Cirelson's bound) cannot be obtained even when the measurement
is performed sequentially first by Alice followed by Bob. In this paper, we
have shown that if von Neumann projection rule is used, quantum bound of CHSH
inequality exceeds it's Cirelson's bound and may also reach its algebraic
maximum four. This thus provide a strong objection regarding the viability of
von Neumann rule as a valid state reduction rule. Further, we pointed out that
the violation of Cirelson's bound occurs due to the injection of additional
quantum non-locality by the act of implementing von Neumann measurement rule.
