Summary: Let \(k\) be a positive integer and let \(D\) be a digraph. A path partition \(\mathcal {P}\) of \(D\) is a set of vertex-disjoint paths which covers \(V(D)\). Its \(k\)-norm is defined as \(\sum _{P \in \mathcal {P}} \min \{|V(P)|, k\}\). A path partition is \(k\)-optimal if its \(k\)-norm is minimum among all path partitions of \(D\). A partial \(k\)-coloring is a collection of \(k\) disjoint stable sets. A partial \(k\)-coloring \(\mathcal {C}\) is orthogonal to a path partition \(\mathcal {P}\) if each path \(P \in \mathcal {P}\) meets \(\min \{|V(P)|,k\}\) distinct sets of \(\mathcal {C}\). C. Berge [Eur. J. Comb. 3, 97–101 (1982; Zbl 0501.05037)] conjectured that every \(k\)-optimal path partition of \(D\) has a partial \(k\)-coloring orthogonal to it. A (path) \(k\)-pack of \(D\) is a collection of at most \(k\) vertex-disjoint paths in \(D\). Its weight is the number of vertices it covers. A \(k\)-pack is optimal if its weight is maximum among all \(k\)-packs of \(D\). A coloring of \(D\) is a partition of \(V(D)\) into stable sets. A \(k\)-pack \(\mathcal {P}\) is orthogonal to a coloring \(\mathcal {C}\) if each set \(C \in \mathcal {C}\) meets \(\min \{|C|, k\}\) paths of \(\mathcal {P}\). R. Aharoni et al. [Pac. J. Math. 118, 249–259 (1985; Zbl 0569.05024)] conjectured that every optimal \(k\)-pack of \(D\) has a coloring orthogonal to it. A digraph \(D\) is semicomplete if every pair of distinct vertices of \(D\) are adjacent. A digraph \(D\) is locally in-semicomplete if, for every vertex \(v \in V(D)\), the in-neighborhood of \(v\) induces a semicomplete digraph. Locally out-semicomplete digraphs are defined similarly. In this paper, we prove Berge’s and Aharoni-Hartman-Hoffman’s conjectures for locally in/out-semicomplete digraphs.