In a smoothly bounded domain
$\Omega \subset \mathbb{R}^n$
,
$n\ge 1$
, this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system
\begin{eqnarray*} \left \{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[5pt] v_t = \Delta v + u - \alpha uv, \end{array} \right . \end{eqnarray*}
with parameter
$\alpha \gt 0$
and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics.
It is shown that there exists
$\delta _\star =\delta _\star (n)\gt 0$
such that for any given
$\alpha \ge \frac{1}{\delta _\star }$
and for any suitably regular initial data satisfying
$v(\cdot, 0)\le \delta _\star$
, this problem admits a unique classical solution that stabilizes to the constant equilibrium
$(\frac{1}{|\Omega |}\int _\Omega u(\cdot, 0), \, \frac{1}{\alpha })$
in the large time limit.