Cyclic difference matrix conjecture for the Fibonacci-like partition construction

Let $q_{i,j}=q_{i,j}^{[m,n]}$ be the non-negative integers obtained in Algorithm; these integers are independent of $m$ and $n$. For each $i\geq 1$, consider the sequence $\{q_{i,j}\}_{j\in\mathbb Z^{\geq 0}}$. Cyclic difference matrix conjecture. For every $i\geq 1$, the sequence $\{q_{i,j}\}_{j\in\mathbb Z^{\geq 0}}$ is a permutation of $\mathbb Z^{\geq 0}$, and for every pair of natural numbers $i_1<i_2$, every integer appears exactly once in the sequence

$$\{q_{i_1,j}-q_{i_2,j}\}_{j\in\mathbb Z^{\geq 0}}.$$

If true, this would give the desired countably infinite analogue of a difference matrix over the infinite cyclic group, whose additive group is the group of integers. The claim is presented as a conjectural generalization of an earlier theorem; no resolution is supplied in the given text.