YES

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

CP(R):
a(___y4) -> b(c(___y4))
a(___y3) -> b(c(___y3))
b(c(___y11)) -> a(___y11)
a(b(c(___y13))) -> b(c(a(___y13)))
c(___y29) -> b(a(c(___y29)))
b(___y40) -> b(a(c(___y40)))
b(a(c(___y47))) -> b(___y47)

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

CR(C): see below.

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