YES

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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