Three-parameter Cartesian-product conjecture for disconnected large bifurcation supports

Let a generic $3$-parameter family have a disconnected large bifurcation support. Three-parameter Cartesian-product conjecture. The family is equivalent either to a Cartesian product of $3$ one-parameter bifurcations, or to a Cartesian product of $2$ bifurcations, one $1$-parameter and the other $2$-parameter. This is the next conjectural case of the proposed decomposition of families according to disconnected support components.