Nonexistence of constant-factor approximation algorithms for directed strengthening inversions

Let $k$ be a positive integer, let $\alpha$ be a constant, and let $D$ be an oriented graph. An $\alpha$-approximation algorithm for a minimization problem returns a solution whose value is at most $\alpha$ times the optimum.

Approximation-hardness conjecture. Unless P=NP, there is no $\alpha$-approximation algorithm for computing $\operatorname{sinv}_k(D)$ or $\operatorname{sinv}'_k(D)$.

This conjecture strengthens the paper's approximation inapproximability results by ruling out polynomial-time constant-factor approximation algorithms for either inversion parameter, for every positive integer $k$.