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(___y1)) -> W(W(___y1))
W(W(___y19)) -> B(I(___y19))
W(B(___y29)) -> W(B(___y29))
F(H(___y56),G(I(___y59))) -> F(G(H(___y56)),I(___y59))
F(G(H(___y74)),I(___y73)) -> F(H(___y74),G(I(___y73)))

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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
W(W(x)) -> W(x)
B(I(x)) -> W(x)
W(B(x)) -> B(x)
F(x,I(y)) -> F(G(x),I(y))
G(x) -> x
CP(R):
W(W(___y97)) -> W(W(___y97))
W(W(___y112)) -> B(I(___y112))
W(B(___y119)) -> W(B(___y119))