Nearly fibered links with genus one

We classify all the $n$-component links in the $3$-sphere that bound a Thurston norm minimizing Seifert surface $\mathrm{\Sigma }$ with Euler characteristic $\chi (\mathrm{\Sigma })=n-2$ and that are nearly fibered, which means that their rank of the maximal (collapsed) Alexander grading $s_{\text{top}}$ of the link Floer homology group $\widehat{HFL}$ is equal to two. In other words, such a link $L$ satisfies $s_{\text{top}}=\frac{n-\chi (\mathrm{\Sigma })}{2}=1$, and in addition $\text{rk}{\widehat{HFL}}_{\ast }(L)[1]=2$ and $\text{rk}{\widehat{HFL}}_{\ast }(L)[s]=0$ for every $s>1$. The proof of the main theorem is inspired by the one of a similar recent result for knots by Baldwin and Sivek; and involves techniques from sutured Floer homology. Furthermore, we also compute the group $\widehat{HFL}$ for each of these links.