Transitivity conjecture for exceptional bases of the Grothendieck group of projective space

Let $K_0(\mathbb{P}_n)$ be the Grothendieck group of coherent sheaves on projective space, equipped with the Euler form

$$\chi(E,F)=\sum_{\nu}(-1)^\nu\dim \operatorname{Ext}^\nu(E,F).$$

An ordered basis $E_0,E_1,\ldots,E_n$ is exceptional if $\chi(E_i,E_j)=0$ for $i>j$ and $\chi(E_i,E_i)=1$ for every $i$. Mutations are the transformations of exceptional bases induced by the braid-group action described in the source, and an isometry of $K_0(\mathbb{P}_n)$ is an automorphism preserving $\chi$.

Transitivity conjecture. The group spanned by mutations of exceptional bases and the isometries of $K_0(\mathbb{P}_n)$ acts transitively on the set of exceptional bases of $K_0(\mathbb{P}_n)$.

This conjecture concerns the classification of exceptional bases up to mutations and Euler-form isometries. The paper presents it as a central conjecture related to the braid-group action; the supplied text does not state a resolution.