Reduced-fiber stratification conjecture for incidence strata

Let $X$ be the variety occurring in the family $\Delta^k(X)$ over $(\mathbb{C}^k)^{\circ}$. For $(m_1,\ldots,m_k),(n_1,\ldots,n_k)\in(\mathbb{C}^k)^{\circ}$, define an equivalence relation by

$$(m_1,\ldots,m_k)\sim(n_1,\ldots,n_k)$$

if

$$\sum_{i\in A}m_i=\sum_{i\in B}m_i\Longleftrightarrow\sum_{i\in A}n_i=\sum_{i\in B}n_i$$

for all $A,B\subset\{1,\ldots,k\}$. Reduced-fiber stratification conjecture. The stratification of $(\mathbb{C}^k)^{\circ}$ induced by this equivalence relation has the property that the preimage in $\Delta^k(X)$ of any equivalence class has reduced fibers over $(\mathbb{C}^k)^{\circ}$. The conjecture is motivated by nonreduced scheme-theoretic fibers caused by point collisions. General reducedness can be obtained after a partition into locally closed sets, but it is unclear whether a partition independent of $X$ exists; the proposed collision-combinatorics stratification is intended to provide such a partition.