Perturbation conjecture for distinct distances in strictly normed planes

Let $\mathbb{M}^2$ be a strictly normed plane and let $S$ be a finite set of $n$ points. Label its points as $S=\{p_1,p_2,\ldots,p_n\}$, and let $<_B$ be an ordering of the pairwise distances such that

$$\|p_i-p_j\|<\|p_k-p_l\|\implies \|p_i-p_j\|<_B\|p_k-p_l\|.$$

Perturbation conjecture. For every $\epsilon>0$, there is a set $S'=\{p'_1,p'_2,\ldots,p'_n\}$ in bijective correspondence with $S$ such that each corresponding pair satisfies $\|p'_i-p_i\|<\epsilon$, all distances between points of $S'$ are distinct, and

$$\|p'_i-p'_j\|<\|p'_k-p'_l\|\iff\|p_i-p_j\|<_B\|p_k-p_l\|.$$

Such a perturbation would allow the maximum-spanning-tree algorithm to be applied directly to point sets with equal distances, while preserving every strict comparison already present among the original distances.