Cyclically distinct composition interpretation for the leafed-cycle cone

Let $\mathcal{C}_n$ be the cone constrained by the Laplacian minor of a leafed $n$-cycle, and let $\lambda=(\lambda_0,\ldots,\lambda_{n-1})$ be an integer point of $\mathcal{C}_n$. For a non-negative integer $m$, consider the integer points with first coordinate $\lambda_0=m$.

Cyclically distinct composition conjecture. The number of integer points $\lambda\in\mathcal{C}_n$ with $\lambda_0=m$ is equal to the number of compositions of $m$ into $n$ parts that are cyclically distinct.

This conjecture would provide a combinatorial interpretation for the coefficients of $\sigma_{\mathcal{C}_n}(q)$.