YES

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

CP(R):
f(g(c(),c())) -> f(g(b(),b()))

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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
f(a()) -> f(g(b(),b()))
c() -> d()
b() -> d()
CP(R):