Simplicial Tutte flow conjecture

Let $d$ be a dimension, let $\boldsymbol{\kappa(d)}$ be a positive integer bound, and let a bridgeless complex of dimension $d$ be a simplicial complex satisfying the paper's definition of bridgelessness. A nowhere-zero $q$-flow is a modular $q$-flow that is nonzero on every facet.

Simplicial Tutte flow conjecture. For every dimension $d$ there exists a number $\kappa(d)<\infty$ such that every bridgeless complex of dimension $d$ has a nowhere-zero $q$-flow for some $q\leq\kappa(d)$.

This is the basic higher-dimensional analogue of Tutte's 5-flow conjecture, replacing the fixed number $5$ by a dimension-dependent finite bound. The paper subsequently seeks linear lower bounds and exponential upper bounds for this quantity.