Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in the set $\{-1, 0, 1\}$. Suppose that $a - \varepsilon t \in \mathbb{Z}_{> 0}^m$ and that $t \not\in \ker \varepsilon$.
Is it true that there exists a sequence of vectors $t = t_1, t_2, \ldots, t_l = 0 \in \mathbb{Z}^k$ with $|t_1| > |t_2| > \ldots > |t_l| = 0$ such that $t_1, \ldots, t_l$ form a lattice path, that is, $t_i$ is obtained from $t_{i - 1}$ by modifying exactly one component by $1$ or $-1$, and such that $a - \varepsilon t_i \in \mathbb{Z}_{> 0}^m$ for all $1 \leq i \leq l$ ?
Or can you provide a counterexample for the above? Or better yet, do you know of any conditions on $\varepsilon$ for which this might be true?