YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
a1() -> b1()
a1() -> c1()
b1() -> b2()
c1() -> c2()
a2() -> b2()
a2() -> c2()
b2() -> b3()
c2() -> c3()
a3() -> b3()
a3() -> c3()
b3() -> b4()
c3() -> c4()
a4() -> b4()
a4() -> c4()
b4() -> b5()
c4() -> c5()
a5() -> b6()
b5() -> b6()
c5() -> b6()

CP(R):
b1() -> c1()
c1() -> b1()
b2() -> c2()
c2() -> b2()
b3() -> c3()
c3() -> b3()
b4() -> c4()
c4() -> b4()

critical-pair-closing system C:
b1() -> b2()
c1() -> c2()
b2() -> b3()
c2() -> c3()
b3() -> b4()
c3() -> c4()
b4() -> b5()
c4() -> c5()
b5() -> b6()
c5() -> b6()

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
b1() -> b2()
c1() -> c2()
b2() -> b3()
c2() -> c3()
b3() -> b4()
c3() -> c4()
b4() -> b5()
c4() -> c5()
b5() -> b6()
c5() -> b6()

CP(R):


critical-pair-closing system C:


CR(C): see below.

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