The wreath-product conjecture for monodromy of reduced irreducible tuples

Let $A$ be a tuple in the setting of Step (1) of Classification, and let $B$ be reduced and irreducible of mixed volume $d$. Write $j$ for the map occurring in that classification, let $\mathop{\rm coker}\nolimits j$ denote its cokernel, and let $G_A$ be the monodromy group associated with $A$. Wreath-product conjecture. The monodromy group $G_A$ equals the wreath product of $\mathop{\rm coker}\nolimits j$ and $S_d$ acting on $\{1,\ldots,d\}$. This conjecture concerns the monodromy of non-reduced tuples; the surrounding discussion explains that reducible systems are more complicated and makes no prediction for the general reducible case.