The unbearable projectivity of being a good moduli space of Gorenstein curves of genus one with non-colliding markings

We classify all open substacks admitting a proper good moduli space inside the stack ${\mathcal{G}}_{1,n}$ of log-canonically polarized Gorenstein curves of genus one with $n$ distinct marked points: they are exactly the stacks ${\bar{\mathcal{M}}}_{1,n}(c)$ defined by Bozlee–Kuo–Neff. Restricting to the case where $c$ is pure of level $m$, we then give a numerical criterion for the projectivity of the good moduli spaces: these turn out to be asymptotically almost never projective as $n$ tends to infinity.