Uniqueness of weak multi-species Nernst–Planck–Poisson solutions in three dimensions

Let $\Omega \subset \mathbb{R}^3$ be a bounded connected open set with $C^2$ boundary, $P \ge 1$ and $z_1, \dots, z_P \in \mathbb{Z}$. The diffusivities $d_i(t, x)$ are measurable and, for each finite $T > 0$, satisfy $0 < d_*(T) \le d_i \le d^*(T) < \infty$ almost everywhere on $Q_T = (0, T) \times \Omega$. The reaction terms $f_i(t, x, y)$ are continuous and bounded on $[0, \infty) \times \overline{\Omega} \times [0, \infty)^P$, continuously differentiable in $y$ with continuous derivatives there, and quasi-positive: $f_i(t, x, y) \ge 0$ whenever $y_i = 0$. Fix $\tau \in L^\infty(\partial\Omega)$, nonnegative and not almost everywhere zero, time-independent $\xi \in L^q(\partial\Omega)$ for some $q > 2$, and nonnegative $c_i^0 \in L^p(\Omega)$ for some finite $p \ge 2$. Consider $$\partial_t c_i - \nabla \cdot \bigl( d_i (\nabla c_i + z_i c_i \nabla \Phi) \bigr) = f_i(t, x, c), \qquad -\Delta \Phi = \sum_{i=1}^P z_i c_i,$$ with no-flux boundary conditions for each $c_i$ and $\partial_\nu \Phi + \tau \Phi = \xi$ on $\partial\Omega$. A weak solution is a pair $(c, \Phi)$ with $c_i \ge 0$ such that for every finite $T > 0$ $$c_i \in C([0, T]; L^1(\Omega)) \cap L^1(0, T; W^{1, 3/2}(\Omega)), \qquad c_i(0) = c_i^0,$$ $$\Phi \in L^\infty(0, T; H^1(\Omega)) \cap L^2(0, T; H^2_{\rm loc}(\Omega)) \cap C([0, T]; L^r(\Omega)) \quad \text{for every } 1 \le r < 6,$$ $F_i = d_i (\nabla c_i + z_i c_i \nabla \Phi) \in L^1(Q_T; \mathbb{R}^3)$, $$\int_{Q_T} \bigl( -c_i \partial_t \psi + F_i \cdot \nabla \psi \bigr) = \int_\Omega c_i^0 \psi(0) + \int_{Q_T} f_i(t, x, c)\, \psi$$ for every smooth $\psi$ on $[0, T] \times \overline{\Omega}$ with $\psi(T) = 0$, and, for almost every $t$, $$\int_\Omega \nabla \Phi(t) \cdot \nabla \varphi + \int_{\partial\Omega} (\tau \Phi(t) - \xi) \varphi = \int_\Omega \Bigl( \sum_{i=1}^P z_i c_i(t) \Bigr) \varphi$$ for every smooth $\varphi$ on $\overline{\Omega}$. Bothe, Fischer, Pierre and Rolland proved that a weak solution in this sense exists for all such data (2014).

Problem. For every such set of data, is there at most one weak solution $(c, \Phi)$?