Global uniqueness of fine-mesh semilinear Galerkin solutions

Let $\Omega \subset \mathbb{R}^d$, $d = 2$ or $3$, be a bounded convex polygon or polyhedron. Let $a_{ij}$ be real Lipschitz functions on $\overline\Omega$ with $\sum_{i,j} a_{ij}(x)\xi_i\xi_j \ge \Lambda |\xi|^2$ for some $\Lambda > 0$, let $b \in L^p(\Omega; \mathbb{R}^d)$ with $p > 2$ if $d = 2$ and $p \ge 3$ if $d = 3$, and $\operatorname{div} b \in L^2(\Omega)$, and fix $u \in L^2(\Omega)$. Let $f \colon \Omega \times \mathbb{R} \to \mathbb{R}$ be a Carathéodory function, nondecreasing in its second variable, with $f(\cdot, 0) \in L^2(\Omega)$ and such that for every $M > 0$ there is $\phi_M \in L^2(\Omega)$ with $|f(x,s) - f(x,t)| \le \phi_M(x)|s - t|$ for almost every $x$ and all $|s|, |t| \le M$; no global growth bound on $f$ is imposed. Take any conforming, shape-regular, quasi-uniform family of simplicial meshes of $\overline\Omega$ with maximal element diameter $h \to 0$, let $V_h \subset H^1_0(\Omega)$ be the continuous piecewise affine functions on the mesh of size $h$, and let $$a(v, w) = \int_\Omega \Bigl( \sum_{i,j} a_{ij}\, \partial_i v\, \partial_j w + (b \cdot \nabla v)\, w \Bigr) dx.$$ The discrete problem is to find $y_h \in V_h$ with $$a(y_h, v_h) + \int_\Omega f(x, y_h)\, v_h \, dx = \int_\Omega u\, v_h \, dx \qquad (v_h \in V_h),$$ with exact integrals. Casas, Mateos and Rösch proved that the corresponding Dirichlet problem has a unique solution $y \in H^2(\Omega) \cap H^1_0(\Omega)$, although the convection term makes it neither monotone nor coercive. For the discrete problem they proved: if a single $\phi \in L^2(\Omega)$ bounds both $|f(x,t)|$ and the Lipschitz constant of $f(x, \cdot)$ on all of $\mathbb{R}$, then it has exactly one solution for all small $h$; and under the assumptions above, for every $M \ge 1 + \|y\|_{C(\overline\Omega)}$ there is $h_M > 0$ such that for $h < h_M$ it has exactly one solution $y_h$ with $\|y_h\|_{C(\overline\Omega)} \le M$, these solutions converge to $y$, and any other family of solutions has $C(\overline\Omega)$-norms tending to infinity.

Problem. Is there $h_0 > 0$, depending on the data and the mesh family, such that for every $h < h_0$ the discrete problem has exactly one solution $y_h \in V_h$? Uniqueness is among all discrete solutions, with no a priori bound in $L^\infty$; a counterexample must keep the coefficients, the nonlinearity and the forcing fixed along arbitrarily fine meshes.