almost surely. One can define the joint law $\mathcal{P}_x^+$ of $(\chi_u:u\in\mathbb{U})$ starting from an Eve cell of initial size $x>0$ and a leaf $\widehat{\ell}$, such that if $\Gamma_n$ is an event measurable with respect to the sigma-field generated by the cells at generations at most $n\geq 0$, then $$\begin{aligned}
        \mathcal{P}_x^+(\Gamma_n)
        =x^{-\omega_+}\mathcal{E}_x\left(\mathds{1}_{\Gamma_n}\sum_{|u|=n+1}\chi_u(0)^{\omega_+}\right).
    \end{aligned}$$ In [@BBCK16], the ancestors of $\widehat{\ell}$ are selected as follows: let $\widehat{\ell}(n+1)\in\mathbb{N}^{n+1}$ be the parent of $\widehat{\ell}$ at generation $n+1$, then its conditional law is $$\begin{aligned}
        \mathcal{P}_x^+\left(\left.\widehat{\ell}(n+1)=v\right|(\chi_u)_{|u|\leq n}\right)
        =\frac{\chi_v(0)^{\omega_+}}{\sum_{|u|=n+1}\chi_u(0)^{\omega_+}},\qquad\forall v\in\mathbb{N}^{n+1}.
    \end{aligned}$$ The law of the spine is again that of a positive self-similar Markov process $(Y^+(t))_{t\geq 0}=(\exp(\eta^+(\tau_t)))_{t\geq 0}$ with index $\alpha$, that we denote by $\mathbb{P}_x^+$ when $Y^+$ starts from $x>0$. Assigning the role of the Eve cell to the spine instead of $\chi_\emptyset$ reorders the genealogy of the cell system, but leaves the law of $\mathbf{X}$ unchanged, see Section 4.2 in [@BBCK16]. We shall henceforth write $\mathcal{P}_x^+$ for the distribution of the reordered cell system where $\chi_\emptyset$ has law $\mathbb{P}_x^+$, and its daughters generate independent growth-fragmentations given the spine, with laws $(\mathcal{P}_y)_{y>0}$ according to their initial sizes.

The Lévy measure $\Pi^+$ of the Lévy process $\eta^+$ is given by $$\begin{aligned}
\label{Equation Pi Levy measure of eta^+}
        \Pi^+(\mathrm{d}y)=e^{\omega_+y}(\Lambda+\widetilde{\Lambda})(\mathrm{d}y),
    