Norm bound for the contingency-table multi-decomposition

Consider the contingency-table problem with an $m\times n$ table, total sum $\Sigma$, row sums $r_k$, and column sums $c_p$. Fixing a row yields the multi-decomposition into the sets $\Omega_{k;a}$ of tables with prescribed $k$-th row; let $\nu$ be the norm of the associated multi-projection $\Pi$.

Contingency-table norm conjecture. For arbitrary table dimensions $m$ and $n$, table sum $\Sigma$, and row and column sums $r_k$ and $c_p$, this multi-decomposition is non-degenerate and

$$\nu\leq 2.$$

The conjecture is motivated by thousands of numerical simulations. The norm is difficult to calculate because the matrices defining the multi-projection lack the regularity present in the earlier examples, and the general contingency-table mixing problem remains challenging.