YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
W(B(x)) -> W(x)
B(I(x)) -> J(x)
W(I(x)) -> W(J(x))

CP(R):
W(J(___y79)) -> W(I(___y79))

critical-pair-closing system C:
W(I(x)) -> W(J(x))

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
W(I(x)) -> W(J(x))

CP(R):


critical-pair-closing system C:


CR(C): see below.

---- CR(R)
---- R is empty.