YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
a(x) -> b(x)
a(b(x)) -> b(a(c(a(x))))
b(x) -> c(x)
c(c(x)) -> x

CP(R):
b(b(___y4)) -> b(a(c(a(___y4))))
b(a(c(a(___y9)))) -> b(b(___y9))
a(c(___y19)) -> b(a(c(a(___y19))))
c(___y31) -> c(___y31)

critical-pair-closing system C:
a(x) -> b(x)
b(x) -> c(x)
c(c(x)) -> x

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
a(x) -> b(x)
b(x) -> c(x)
c(c(x)) -> x
CP(R):
c(___y49) -> c(___y49)