Summary: This article is concerned with the model \[ \begin{aligned} u_t & =\Delta u-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\quad x\in \Omega,\; t>0,\\ 0 & =\Delta v+\alpha u-\beta v,\quad x\in\Omega,\; t>0,\\ 0 & =\Delta w+\gamma u-\delta w,\quad x\in\Omega,\; t>0\end{aligned} \] with homogeneous Neumann boundary conditions in a bounded domain \(\Omega\subset\mathbb R^n\) \((n=2,3)\). Under the critical condition \(\chi \alpha-\xi \gamma=0\), we show that the system possesses a unique global solution that is uniformly bounded in time. Moreover, when \(n=2\), by some appropriate smallness conditions on the initial data, we assert that this solution converges to \((\bar{u}_0, \frac{\alpha}{\beta}\bar{u}_0, \frac{\gamma}{\delta}\bar{u}_0)\) exponentially, where \(\bar{u}_0:=\frac{1}{|\Omega|}\int_{\Omega}u_0\).