YES

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

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

critical-pair-closing system C:
a(x) -> 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) -> x
b(x) -> c(x)
c(c(x)) -> x
CP(R):
c(___y49) -> c(___y49)