Weierstrass points on the Drinfeld modular curve $X_0(\mathfrak{p})$

Consider the Drinfeld modular curve $X_0(\mathfrak{p})$ for $\mathfrak{p}$ a prime ideal of ${\mathbb{F}}_q[T]$. It was previously known that if $j$ is the $j$-invariant of a Weierstrass point of $X_0(\mathfrak{p})$, then the reduction of $j$ modulo $\mathfrak{p}$ is a supersingular $j$-invariant. In this paper we show the converse: Every supersingular $j$-invariant is the reduction modulo $\mathfrak{p}$ of the $j$-invariant of a Weierstrass point of $X_0(\mathfrak{p})$.