YES

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

CP(R):
f(j(___y32,___y32),g(a())) -> f(j(___y32,___y32),g(b()))

critical-pair-closing system C:
b() -> a()

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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