at rate $p$, while the opposite occurs at rate $q$. We denote by $(h^\zeta(t,\cdot),t\ge 0)$ the associated Markov process starting from some initial configuration $\zeta \in \Omega_{N,k}$.\
It will be convenient to denote by $\wedge$ the maximal height function: $$\wedge(x) = x \wedge (2k-x)\;,\quad x\in{\llbracket}1,N{\rrbracket}\;,$$ and by $\vee$ the minimal height function: $$\vee(x) = (-x)\vee(x-2N+2k)\;,\quad x\in{\llbracket}1,N{\rrbracket}\;.$$ Though the dependence on $k$ is implicit in the notations $\wedge,\vee$, this will never raise any confusion as the value $k$ will be clear from the context.

It is possible to construct simultaneously on a same probability space and in a Markovian fashion, the height function processes $(h_t^\zeta,t\ge 0)$ starting from all initial conditions $\zeta\in\cup_k \Omega_{N,k}$ and such that the following monotonicity property is satisfied for all $k$ and all $\zeta,\zeta' \in \Omega_{N,k}$: $$\label{eq:grand}
\zeta \le \zeta' \Rightarrow h_t^\zeta \le h_t^{\zeta'}\;,\quad \forall t\ge 0\;.$$ Here, $\zeta\le \zeta'$ simply means $\zeta(x) \le \zeta'(x)$ for all $x\in{\llbracket}0,N{\rrbracket}$. We call such a construction a monotone Markovian grand coupling, and we denote by ${\mathbb{P}}$ the corresponding probability distribution. The existence of such a grand coupling is classical, see for instance [@LabLac16 Proposition 4]. In a portion of our proof, we require to use a specific grand coupling which 