YES

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

CP(R):
b() -> f(c())
f(c()) -> b()
f(d()) -> b()
f(d()) -> f(c())
f(c()) -> b()

critical-pair-closing system C:
f(d()) -> b()
f(c()) -> b()

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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