Direction-flip conjecture for oriented paths

For $k\geq 3$, an oriented path with one direction flip is a path with $k$ edges whose directions are consecutive in one direction and then consecutive in the opposite direction, written $\rightarrow\dots \rightarrow \leftarrow\dots \leftarrow$. An oriented path is TAS if it has the tournament anti-Sidorenko property.

Direction-flip conjecture. For all $k\geq 3$, an oriented path with $k$ edges and one direction flip ($\rightarrow\dots \rightarrow \leftarrow\dots \leftarrow$) is TAS.

The paper proves that the six-edge path with the displayed direction pattern is neither TAS nor TS, but this conjecture concerns the broader one-flip family; the apparent tension should be checked against the exact placement of the flip and the preceding example.