YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
W(W(x)) -> W(x)
B(I(x)) -> W(x)
W(B(x)) -> B(x)
F(H(x),y) -> F(H(x),G(y))
F(x,I(y)) -> F(G(x),I(y))
G(x) -> x

CP(R):
W(W(___y3361)) -> W(W(___y3361))
W(W(___y3379)) -> B(I(___y3379))
W(B(___y3389)) -> W(B(___y3389))
F(H(___y3416),G(I(___y3419))) -> F(G(H(___y3416)),I(___y3419))
F(G(H(___y3434)),I(___y3433)) -> F(H(___y3434),G(I(___y3433)))

critical-pair-closing system C:
W(W(x)) -> W(x)
B(I(x)) -> W(x)
G(x) -> x

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
W(W(x)) -> W(x)
B(I(x)) -> W(x)
G(x) -> x

CP(R):
W(W(___y3457)) -> W(W(___y3457))

critical-pair-closing system C:


CR(C): see below.

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