Monotonicity conjecture for the positive margins property

Let $G$ be a graph, let $V$ be its vertex set, and let $d=(d_v)_{v\in V}$ and $d'=(d'_v)_{v\in V}$ be cardinality vectors. The pair $(G,d)$ has the positive margins (or interior point) property when the corresponding model has that property.

Monotonicity conjecture. Assume that $(G,d)$ does not have the positive margins (or interior point) property. If $d'_v\ge d_v$ for all $v\in V$, then $(G,d')$ does not have the positive margins (or interior point) property either.

The conjecture proposes that increasing the cardinalities cannot restore the positive margins or interior point property. The paper notes that a similar phenomenon occurs for Markov bases, but gives no resolution of this conjecture.