Conjecture on reconstruction from transposition-neighborhoods

Let $G=\operatorname{Sym}_n$ and let $T$ be the set of transpositions, so that $\operatorname{Sym}_n(T)$ is the corresponding Cayley graph, $B_r(\sigma)$ denotes the radius-$r$ ball about $\sigma$, and $N(\operatorname{Sym}_n(T),r)$ and $N_2(\operatorname{Sym}_n(T),2)$ are the reconstruction parameters defined in the paper. For any $\pi\in (1^{n-3}\,3^1)^G$, any $r\geq 1$, and $n\geq 2r+1$, the reconstruction conjecture asserts that

$$N(\operatorname{Sym}_n(T),r)=N_2(\operatorname{Sym}_n(T),2)=\lvert B_r(I)\cap B_r(\pi)\rvert.$$

This generalizes the preceding theorem and corollary on reconstruction of permutations from transposition-neighborhoods, predicting that the relevant reconstruction number is independent of the chosen element of the indicated conjugacy class and equals the intersection size of the two radius-$r$ balls. The parser supplies no evidence that the conjecture has been resolved.