Uniqueness of very free curve components in large nef classes

Let $X$ be a smooth Fano threefold. Write $\operatorname{Nef}_1(X)_{\mathbb{Z}}$ for the integral nef curve classes, and let $\operatorname{Mor}(\mathbb{P}^1,X,\alpha)$ denote the space of morphisms of class $\alpha$. Uniqueness conjecture for very free components. There exists $\tau \in \operatorname{Nef}_1(X)_{\mathbb{Z}}$ such that for all $\alpha \in \tau + \operatorname{Nef}_1(X)_{\mathbb{Z}}$, there is at most one component of $\operatorname{Mor}(\mathbb{P}^1, X, \alpha)$ that generically parameterizes very free curves. This is presented as a reformulation of Geometric Manin's Conjecture and is intended to follow from the paper's main theorem in the stated setting; the supplied material does not establish the unrestricted claim as a general theorem.