Facet-connectivity reduction conjecture for simplicial flows

Fix a positive integer $\kappa$. Let $d$ be the dimension and let $k$ satisfy

$$d+2\geq k>2.$$

A complex is $k$-facet-connected according to the paper's definition, and a nowhere-zero $q$-flow is a flow nonzero on every facet.

Facet-connectivity reduction conjecture. If every $k$-facet-connected complex $\Delta$ has a $q$-flow for some $q<\kappa$, then every $(k-1)$-facet-connected complex $\Delta$ also has a nowhere-zero $q$-flow for some $q<\kappa$.

The author proposes this as a final conjectural reduction: iterating it would reduce the finiteness claim for $\kappa(d)$ to the stronger facet-connectivity cases. No resolution is given in the supplied text.