YES

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

CP(R):
f(f(g(f(___y1)))) -> f(g(f(f(___y1))))

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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
f(f(x)) -> f(g(f(x)))
CP(R):
f(f(g(f(___y3)))) -> f(g(f(f(___y3))))