It was shown by the authors that two one-dimensional probability measures in the convex order admit a martingale coupling with respect to which the integral of $\vert x-y\vert$ is smaller than twice their $\mathcal W_1$-distance (Wasserstein distance with index $1$). We showed that replacing $\vert x-y\vert$ and $\mathcal W_1$ respectively with $\vert x-y\vert^ρ$ and $\mathcal W_ρ^ρ$ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing $\mathcal W_ρ^ρ$ with the product of $\mathcal W_ρ$ times the centred $ρ$-th moment of the second marginal to the power $ρ-1$. Then we study the generalisation of this new stability inequality to higher dimension.