Baker–Chen–Li–Qian refinement for cyclic V-shaped permutations

Let $n$ be a positive integer. A cyclic V-shaped permutation of $[n]$ is a permutation with a unique local minimum that is also a cycle, and let the minimum occur at position $k$. Consider subsets of $[n-1]$ having cardinality $k-1$ and whose elements sum to $1\bmod n$. Baker–Chen–Li–Qian's refinement. The number of cyclic V-shaped permutations of $[n]$ whose minimum is at position $k$ equals the number of $(k-1)$-element subsets of $[n-1]$ whose elements sum to $1\bmod n$. This conjecture refines the known equality between the total number of these cyclic V-shaped permutations and the number of primitive necklaces of length $n$ with an odd number of ones, both enumerated by the relevant Möbius-sum formula. The source presents the refinement as a conjecture and gives no resolution.