Kalkowski–Karoński–Pfender's twin-free hypergraph conjecture

Let $r\geq 3$. Define

$$\mathfrak{B}^{r}=\{\mathcal{H}:\mathcal{H}\text{ is an }r\text{-uniform hypergraph and }\chi^{e}(\mathcal{H})>2\}.$$

Two vertices are twins when the sets of edges containing them are equal, and a hypergraph is twin-free when it contains no twins. Kalkowski–Karoński–Pfender's twin-free hypergraph conjecture. There is no twin-free hypergraph in $\mathfrak{B}^{r}$. The conjecture says that every twin-free $r$-uniform hypergraph admits a neighbor sum distinguishing edge weighting using weights 1 and 2. The source gives examples in $\mathfrak{B}^{r}$ that have twins, but does not resolve whether a twin-free example exists.