YES

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

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

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

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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