this result by terms of order $h_x^2/\alpha^2\,p_\text{F}^2$ in Eq. (\[DL\_strong\_SOC\]).

The $q$ expansion of the self-consistency relation Eq. (\[DL\_strong\_SOC\]) exhibits a linear term in $q$ which leads to the emergence of an inhomogeneous superconducting state at wave-vector: $$\hbar q_0 = \frac{3\,h_x^2\,h_z}{4\,\alpha\,\mu^2}\; ,
    \label{q0_strong}$$ where $q_0$ is estimated by maximizing Eq. (\[DL\_strong\_SOC\]) with respect to $q$. This expression of $q_0$ is only valid for large values of $\alpha$, and we have restored the factor $\hbar$. The scaling in $h_x$ and $h_z$ is similar to the one for weak spin orbit, see Eq. (\[q0\_weak\]).

Introducing the parameter $\tilde{\alpha}$ commonly used in the experimental literature, this expressions reads : $$q_0 = \frac{3\,h_x^2\,h_z}{4\,\tilde{\alpha}\,\mu^2}\; .
    \label{q0_strong_eval}$$ For the values $\tilde{\alpha} = 300$ meV.A of the SO coupling, $h_x=h_z=0.5$ meV of the Zeeman components, and $\mu = 1$ eV for the Fermi energy, this formula leads to a very small $q_0$ which justifies [*a posteriori*]{} the expansion in $q$.

\[subsec:numerics\] Numerical results and phase diagrams
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We now solve the self-consistency relation Eq. (\[app:SCE\_fin\]) for arbitrary values of the wave-vector $q$, i.e. without doing any expansion in parameter $q$. We solve numerically the self-consistency equation Eq. (\[app:SCE\_fin\]) in the following way. First, the temperature is extracted 