Conjecture on the classification of varieties with full ramification locus

Let $X$ be a smooth complex variety of dimension $n\ge 2$. Let $L$ be an ample line bundle on $X$, spanned by a subspace $V\subseteq H^0(L)$ with $\dim(|V|)\ge n+1$. Let $\mathcal{R}(X,V)$ denote the ramification locus associated with the linear system $V$. Classification conjecture. If

$$\mathcal{R}(X,V)=X,$$

then $(X,L)$ is one of $(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(2))$, $(\mathbb{P}^{1}\times\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1))$, or a scroll over a smooth curve. The examples preceding the conjecture motivate this proposed classification of ample spanned line bundles whose ramification locus is the whole variety; the statement is presented as a conjecture, and no resolution is supplied in the source.