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(___y420) -> c(b(b(a(a(___y420)))))
c(b(b(a(a(___y425))))) -> b(___y425)
a(c(___y435)) -> c(b(b(a(a(___y435)))))
c(___y447) -> c(___y447)

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

CR(C): see below.

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

CP(R):
c(___y465) -> c(___y465)

critical-pair-closing system C:


CR(C): see below.

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