to an accuracy of better than $1$% at low momenta. This is crucial for example for hadronic flavour-conserving parity violation since it considerably extends the number of targets and observables [@Griesshammer:2010nd].

Based on the asymptotic off-shell amplitudes, Refs. [@Bedaque:2002yg; @improve3body] proposed that $H_2$ is only suppressed by $Q^2$ relative to LO, i.e. that calculations at [N${}^{2}$LO]{} or on the $10$%-level do already need one additional $3N$ datum as input. In Ref. [@effrange], this was confirmed and extended to a general scheme to find the order at which any given $3N$ interaction starts contributing. The argument analyses perturbations to the asymptotic form of the LO integral equation. It is not immediately transparent, as witnessed by a subsequent claim that a $k$-dependent $3N$ interaction enters not earlier than [N${}^{3}$LO]{} [@Platter:2006ev]. Upon closer inspection, it was later refuted [@Ji:2012nj].

Refs. [@Bedaque:2002yg; @improve3body] also supplied numerical evidence from solutions of the Faddeev equations in momentum space with a step-function cutoff: a double-logarithmic plot of eq.  for the inverse $K$ matrix, ${\mathcal{O}}=k\cot\delta$ at $\Lambda_1=900\;{\ensuremath{\mathrm{MeV}}}$ and $\Lambda_2=200\;{\ensuremath{\mathrm{MeV}}}$, both well above the breakdown scale ${\overline{\Lambda}_{\slashed{\pi}}}\approx{\ensuremath{m_\pi}}$ of [EFT($\slashed{\pi}$)]{}. A slight variant is reproduced here as Fig. \[fig:nonLepage\]. The cutoff dependence decreases order-by-order as expected when the theory is 