YES

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

CP(R):
f(s(___y121)) -> h(g(___y121),g(___y121))

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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