YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
f(g(x,a(),b())) -> x
g(f(h(c(),d())),x,y) -> h(k1(x),k2(y))
k1(a()) -> c()
k2(b()) -> d()
f(h(k1(a()),k2(b()))) -> f(h(c(),d()))
f(h(c(),k2(b()))) -> f(h(c(),d()))
f(h(k1(a()),d())) -> f(h(c(),d()))

CP(R):
f(h(k1(a()),k2(b()))) -> f(h(c(),d()))
f(h(c(),k2(b()))) -> f(h(c(),d()))
f(h(c(),d())) -> f(h(c(),d()))
f(h(k1(a()),d())) -> f(h(c(),d()))
f(h(c(),d())) -> f(h(c(),d()))

critical-pair-closing system C:
k1(a()) -> c()
k2(b()) -> d()

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
k1(a()) -> c()
k2(b()) -> d()

CP(R):


critical-pair-closing system C:


CR(C): see below.

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