Conditional negative association for partitioned urn indicators

Let $T_0\cup T_1\cup\cdots\cup T_s$ be a partition of $[m]\times[n]$, and let $a_r,b_r\in\mathbb{N}$ for $r=1,\ldots,s$. For the indicator variables $xi_{ij}$ defined in the paper, write

$$\xi(T)=\sum_{(i,j)\in T}\xi_{ij}.$$

Partitioned-urn conjecture. The variables $xi_{ij}$ are negatively associated conditional on

$$\{\xi(T_r)\in[a_r,b_r]~\forall r\in[s]\}.$$

The source presents this as a possible considerable strengthening of conditional negative association for generalized threshold urn measures; no resolution is supplied.