Weaker form of Shelah's categoricity conjecture for accessible categories

Let $\mathcal{A}$ be a nice accessible category with directed colimits having just one model of presentability rank $\kappa$. Weaker conjecture. Then $\mathcal{A}$ must be categorical in unboundedly many cardinals higher than $\kappa$. This is explicitly presented as a weaker version of Shelah's categoricity conjecture. The source gives no resolution of this weaker assertion.