Automatic conservation laws for renormalized nonlinear diffusion

Let $\Omega \subset \mathbb{R}^d$ be a bounded smooth domain, $N \ge 2$, $d_i > 0$ and $m_i \in (0, 2)$ for $i = 1, \dots, N$, with $m_i \ne 1$ for at least one $i$. Let $f : [0, \infty)^N \to \mathbb{R}^N$ be locally Lipschitz and quasi-positive ($f_i(z) \ge 0$ whenever $z_i = 0$), and assume that for some constants $C \ge 0$ and $\mu_i \in \mathbb{R}$ $$\sum_i f_i(z) (\log z_i + \mu_i) \le C \sum_i (1 + z_i \log z_i) \qquad (z \in (0, \infty)^N).$$ Let $u_0 \ge 0$ have finite entropy, $\sum_i \int_\Omega u_{i,0} (1 + |\log u_{i,0}|) < \infty$ with $0 \cdot |\log 0| = 0$. Consider nonnegative $u = (u_1, \dots, u_N)$ with $u_i \in L^\infty(0, T; L^1(\Omega))$ and $u_i^{m_i/2} \in L^2(0, T; H^1(\Omega))$ for every finite $T$, satisfying the renormalized form of $\partial_t u_i = d_i \Delta u_i^{m_i} + f_i(u)$ with no-flux boundary conditions: for every smooth $\xi$ with compactly supported $D\xi$, every smooth $\psi$ on $\overline{\Omega} \times [0, \infty)$ and almost every $T > 0$, $$\begin{aligned} \int_\Omega \xi(u(T)) \psi(T) - \int_\Omega \xi(u_0) \psi(0) - \int_0^T \!\! \int_\Omega \xi(u) \partial_t \psi ={}& -\sum_{i,j} d_i m_i \int_0^T \!\! \int_\Omega \psi\, \xi_{ij}(u)\, u_i^{m_i - 1} \nabla u_i \cdot \nabla u_j \\ & - \sum_i d_i m_i \int_0^T \!\! \int_\Omega \xi_i(u)\, u_i^{m_i - 1} \nabla u_i \cdot \nabla \psi + \sum_i \int_0^T \!\! \int_\Omega \xi_i(u) f_i(u) \psi, \end{aligned}$$ where $\xi_i$ and $\xi_{ij}$ are partial derivatives of $\xi$, the flux is read as $u_i^{m_i - 1} \nabla u_i = (2/m_i)\, u_i^{m_i/2} \nabla u_i^{m_i/2}$ and the gradient product as $$u_i^{m_i - 1} \nabla u_i \cdot \nabla u_j = \frac{4}{m_i m_j}\, u_i^{m_i/2} u_j^{1 - m_j/2}\, \nabla u_i^{m_i/2} \cdot \nabla u_j^{m_j/2}.$$ Fellner, Fischer, Kniely and Tang proved that global solutions of this identity exist which also satisfy an entropy inequality and the conservation laws below (2023). For linear diffusion, $m_i = 1$ for all $i$, Fischer showed that the entropy inequality and the conservation laws follow from the renormalized identity (2017), and for bounded solutions they can be proved directly.

Problem. Does this identity alone imply, for every $q \in \mathbb{R}^N$ with $q \cdot f(z) = 0$ for all $z \ge 0$, that $$\sum_i q_i \int_\Omega u_i(t) = \sum_i q_i \int_\Omega u_{i,0} \qquad \text{for almost every } t > 0?$$ Neither this conservation law nor an entropy inequality is assumed as part of the definition of a solution.