Hassett's weighted blow-up conjecture for unordered points on the projective line

Let $n$ be a positive integer and let $\frac{1}{n}+\frac{}{}\epsilon$ be the common weight on $2n$ marked points, with the weighted moduli space $\overline{\mathcal{M}}_{0,(\frac{1}{n}+\epsilon,\ldots,\frac{1}{n}+\epsilon)}$. Let $\widetilde{M}_{0,2n}$ be the coarse moduli space of

$$[\overline{\mathcal{M}}_{0,(\frac{1}{n}+\epsilon,\ldots,\frac{1}{n}+\epsilon)}/S_{2n}],$$

where $S_{2n}$ permutes the marked points, and let

$$\mathbb{P}(\operatorname{H}^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(2n)))/\!\!/\operatorname{PGL}_2$$

be the GIT moduli space of $2n$ unordered points on $\mathbb{P}^1$, linearized via $\mathcal{O}(2)$. Hassett's conjecture. There is a map

$$\widetilde{M}_{0,2n}\to \mathbb{P}(\operatorname{H}^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(2n)))/\!\!/\operatorname{PGL}_2$$

which is a weighted blow-up. The conjecture identifies the birational geometry of the moduli space of unordered weighted stable curves with the GIT compactification of unordered points on $\mathbb{P}^1$; the stated result is proven by showing the corresponding statement for ordered points and constructing the induced map.