Bang-Jensen and Gutin's longest path algorithm conjecture for semicomplete digraphs

Let $D$ be a semicomplete digraph, and let $x,y$ be distinct vertices of $D$. A longest $(x,y)$-path is an $(x,y)$-path containing the maximum possible number of arcs. Longest path algorithm conjecture. There exists a polynomial algorithm for finding a longest $(x,y)$-path when the input is a semicomplete digraph $D$ and $x,y$ are distinct vertices of $D$. The complexity of finding such a path is stated to be open, even for semicomplete digraphs; the conjecture asks for a polynomial-time algorithm.