Northcott conjecture for stacky curves with coarse space $\mathbb{P}^1$

For a stacky curve $\mathfrak{X}=\mathfrak{X}(\mathbb{P}^1:({\mathbf a},{\mathbf m}))$, let $\gamma(\mathfrak{X})$ be the infimum of the real numbers $\delta$ for which the modified height $\mathcal{H}_{({\mathbf a},{\mathbf m})}^\delta$ has the Northcott property on $\mathfrak{X}$. Northcott conjecture. For all $({\mathbf a},{\mathbf m})$,

$$\gamma(\mathfrak{X})=\min\{\delta({\mathbf m}),0\}.$$

The conjecture concerns the threshold at which the modified height acquires the Northcott property for stacky curves with coarse space $\mathbb{P}^1$; the source does not specify its resolution status.