YES

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

CP(R):


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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
a(a(b(b(x)))) -> b(b(b(b(b(a(a(a(a(a(x))))))))))
CP(R):