YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
f(x) -> g(k(x))
f(x) -> a()
g(x) -> a()
k(a()) -> k(k(a()))

CP(R):
g(k(___y3)) -> a()
a() -> g(k(___y8))

critical-pair-closing system C:
f(x) -> g(k(x))
g(x) -> a()

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
f(x) -> g(k(x))
g(x) -> a()
CP(R):