The converse to the surplus condition criterion for point configurations

Let $r,n$ be integers, let $[n]=\{1,\ldots,n\}$, and let $\mathcal{I}=(I_1,\ldots,I_k)$ be a collection of subsets of $[n]$. Let $d_{r,n}(\mathcal{I})$ denote the degree associated with the forgetful map for this collection, and say that $\mathcal{I}$ satisfies the surplus condition when, for every nonempty subset $J\subseteq [k]$, one has

$$\left|\bigcup_{j\in J}I_j\right|\geq |J|+r+1.$$

The converse to the surplus condition criterion. One has

$$d_{r,n}(\mathcal{I})\neq 0$$

if and only if $\mathcal{I}$ satisfies the surplus condition.

The proposition preceding this conjecture proves the forward implication from nonzero degree to the surplus condition. The conjecture asserts the converse, namely that the surplus condition is also sufficient for nonvanishing of the degree.