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(___y291) -> h(f(___y291))
h(g(___y293)) -> h(g(___y293))
h(f(___y297)) -> g(___y297)
h(h(f(___y301))) -> h(g(___y301))

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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