YES

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

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

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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