YES

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

CP(R):
f(a()) -> a()
f(f(a())) -> a()
f(a()) -> a()
f(f(a())) -> a()
f(f(a())) -> a()
f(a()) -> a()
g(a()) -> a()
f(f(a())) -> a()
f(f(f(a()))) -> a()
f(f(g(a()))) -> f(a())
f(f(g(a()))) -> f(a())

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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