Minimal-volume conjecture for meridian type vector fields

Let $M^\star$ be the punctured 2-sphere under consideration, and for each positive integer $k$ let $X_{\mathrm{m},k}$ denote the meridian type vector field whose relative homology class is indexed by $k$. Meridian minimal-volume conjecture. For each $k\in\mathbb{Z}^+$, the meridian type vector field $X_{\mathrm{m},k}$ realizes minimal volume in its relative homology class. The cases $k=0,1$ are known, while minimality for the remaining meridian classes is conjectural and depends on the topology of the domain and the vector fields.