YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
or(true(),true()) -> true()
or(x,y) -> or(y,x)

CP(R):
true() -> or(true(),true())
or(true(),true()) -> true()

critical-pair-closing system C:
or(true(),true()) -> true()

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
or(true(),true()) -> true()

CP(R):


critical-pair-closing system C:


CR(C): see below.

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