Jaeger–Linial–Payan–Tarzi additive bases conjecture for finite vector spaces

Let $p$ be a prime, let $\mathbb{Z}_p^n$ be the $n$-dimensional vector space over $\mathbb{Z}_p$, and let an additive basis be a multiset $B$ of elements of $\mathbb{Z}_p^n$ such that

$$\mathcal{S}(B)=\left\{\sum_{b\in S}b\mid S\subseteq B\right\}=\mathbb{Z}_p^n.$$

A basis is a basis of the vector space $\mathbb{Z}_p^n$.

Jaeger–Linial–Payan–Tarzi conjecture. For every prime $p$, there exists a constant $k_p$ such that the union, with repetitions, of any $k_p$ bases for $\mathbb{Z}_p^n$ forms an additive basis.

This conjecture would generalize important results concerning nowhere-zero flows; in particular, the case $p=3$ would imply the weak $3$-flow conjecture. The source does not provide evidence of a resolution, so the conjecture is recorded as open.