Uniqueness of binary phylogenetic trees from their $A_k$ alignments

Let $k\in\mathbb{N}_{\geq 3}$, let $X$ be a taxon set, and let $T$ and $\widetilde{T}$ be binary phylogenetic trees on $X$. For a binary phylogenetic tree $T$, let $A_k(T)$ denote the set of binary characters requiring precisely $k$ nucleotide substitutions on $T$. Uniqueness conjecture. If $|X|\geq 2k+3$ and $T\not\cong\widetilde{T}$, then

$$A_k(T)\neq A_k(\widetilde{T}).$$

The claim would close the gap between the known counterexamples through $2k+2$ leaves and the paper's positive result for at least $4k$ leaves. It remains open in the source.