The wrapped Fukaya category as localization along continuation maps

Let $\mathcal{S}$ be an abstract wrapped Floer setup, let $\mathcal{F}^{\text{pre}}_{\mathcal{S}}$ be its pre-wrapped Fukaya $A_\infty$-category, let $\widetilde{C}$ be the set of continuation maps, and let $\mathcal{W}_{P_{\text{univ}}(\mathcal{S}),\widetilde{C}}$ denote the associated wrapped Fukaya category. For any $A_\infty$-category $\mathcal{C}$ and suitably defined $\operatorname{Fun}(\mathcal{F}^{\text{pre}}_{\mathcal{S}},\mathcal{C})$, Localization conjecture. the restriction functor

$$\operatorname{Fun}(\mathcal{W}_{P_{\text{univ}}(\mathcal{S}),\widetilde{C}},\mathcal{C})\to\operatorname{Fun}(\mathcal{F}^{\text{pre}}_{\mathcal{S}},\mathcal{C})$$

is fully faithful, and its essential image consists of those functors that send continuation maps to isomorphisms in $\mathcal{C}$. This asserts that the wrapped Fukaya category is characterized as the $\infty$-categorical localization of the pre-wrapped category along continuation maps; the paper states that it gives an affirmative answer to this conjecture.