The small Fuss-Schröder path–sparse noncrossing partition bijection conjecture

Let $k$ and $n$ be positive integers, and let $\lambda=\lambda_1\lambda_2\ldots\lambda_\ell$ be a type with $\lvert\lambda\rvert$ denoting its size. A small $(k,k)$-Fuss-Schröder path is a path of type $\lambda$ and length $n$, and a connected sparse noncrossing partition is a sparse noncrossing partition with one connected component. The small-path bijection conjecture. The set of small $(k,k)$-Fuss-Schröder paths of type $\lambda$ and length $n$ is in bijection with the set of connected sparse noncrossing partitions of $[2(k+1)n+1]$ such that

1. the arc type is $((k+1)\lambda_1,(k+1)\lambda_2,\dots,(k+1)\lambda_\ell,(k+1)^{n-\lvert\lambda\rvert})$;
2. for $0\leq i\leq n-1$, the set of $(i(k+1)+2)$th blocks consists of $n-\lvert\lambda\rvert$ blocks of arc type $k+1$ and $\lvert\lambda\rvert$ singleton blocks; and
3. the set of the last $t(k+1)$ blocks has at least $t(k-1)+1$ singleton blocks for $t\geq1$.

This conjecture is presented as the connected-partition counterpart of the preceding large-path conjecture: when the second connected component in the large case is a singleton block, the corresponding path is small. The supplied text gives no evidence that this conjecture has been proved or refuted.