The fixed-arc deletion algorithm conjecture for semicomplete digraphs

Let $k$ be a fixed positive integer. A semicomplete digraph is a directed graph in which every pair of distinct vertices is joined by at least one arc. A spanning eulerian subdigraph is a subdigraph containing every vertex in which each vertex has equal indegree and outdegree. For a semicomplete digraph $D=(V,A)$ and a set $A'\subset A$, write $D\setminus A'$ for the digraph obtained by deleting the arcs in $A'$.

Fixed-arc deletion algorithm conjecture. For each fixed positive integer $k$, there exists a polynomial-time algorithm which, given a semicomplete digraph $D=(V,A)$ and $A'\subset A$ with $|A'|=k$, decides whether $D\setminus A'$ has a spanning eulerian subdigraph.

The paper gives a polynomial-time algorithm when one arc is deleted and conjectures that this tractability extends to every fixed number of deleted arcs. The general fixed-$k$ case remains open.