Nonexistence of vector bundles realizing a multiple of stacky height on elliptic curves

Let $k$ be a field of characteristic $3$, let $\overline{\mathscr M}_{1,1}$ be the Deligne–Mumford stack of stable elliptic curves over $k$, and let $\height_\mathscr W(x)$ denote the height associated with a vector bundle $\mathscr W$ on $\overline{\mathscr M}_{1,1}$.

Height-multiple nonexistence conjecture. There is no vector bundle $\mathscr W$ on $\overline{\mathscr M}_{1,1}$ for which there exists an integer $n$ such that

$$n\height(x)=\height_\mathscr W(x).$$

The preceding theorem establishes the analogous nonexistence result with $n=1$ for points over finite extensions of $k(t)$, while the existence of such a vector bundle and integer remains open for arbitrary multiples of the height in characteristic $3$.