YES

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

CP(R):
g(___y3) -> h(f(___y3))
h(g(___y5)) -> h(g(___y5))
h(f(___y9)) -> g(___y9)
h(h(f(___y13))) -> h(g(___y13))

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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
f(x) -> g(x)
h(f(x)) -> h(g(x))
g(x) -> h(g(x))
CP(R):
h(g(___y35)) -> h(g(___y35))