The spanning eulerian subdigraph avoidance conjecture for semicomplete digraphs

Let $k$ be a non-negative integer. A semicomplete digraph is a directed graph in which every pair of distinct vertices is joined by at least one arc. A digraph is $(k+1)$-arc-strong if deleting fewer than $k+1$ arcs leaves it strongly connected. A spanning eulerian subdigraph is a subdigraph containing every vertex in which each vertex has equal indegree and outdegree.

Spanning eulerian subdigraph avoidance conjecture. Every $(k+1)$-arc-strong semicomplete digraph $D$ has a spanning eulerian subdigraph that avoids any prescribed set of $k$ arcs.

The paper proves the existence of a finite connectivity bound, with an upper bound of $(k+1)^2/4+1$, while a degree-based obstruction gives the lower bound $k+1$. The conjecture that the lower bound always suffices remains open.