Hardness conjecture for bounded-size inversion approximation

Let $p\geq 3$ and $k\geq 1$ be integers. For a digraph $D$, let $\operatorname{sinv}_{k}^{\leq p}(D)$ denote the optimization problem of making $D$ $k$-arc-strong using inversions of size at most $p$.

Bounded-size inversion approximation hardness conjecture. There exists $\varepsilon>0$ such that, unless $P=NP$, for every $p\geq 3$ and every $k\geq 1$, no polynomial-time $(1+\varepsilon)$-approximation algorithm for $\operatorname{sinv}_{k}^{\leq p}$ in digraphs exists.

The conjecture asserts that increasing the allowed inversion size does not make the approximation problem easier. It is presented as an open conjecture in the source.