YES

1 decompositions
 #0 -----------
   1: W(B(x)) -> W(x)
   2: B(I(x)) -> J(x)
   3: W(I(x)) -> W(J(x))

@Strongly Commuting
--- R
   1: W(B(x)) -> W(x)
   2: B(I(x)) -> J(x)
   3: W(I(x)) -> W(J(x))

--- S
   1: W(B(x)) -> W(x)
   2: B(I(x)) -> J(x)
   3: W(I(x)) -> W(J(x))