We consider the solution of the torsion problem $-��u=1$ in $��$ and $u=0$ on $\partial ��$. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_��$ is constant on $\partial ��$, then $��$ must be a ball. In a recent paper, it has been conjectured that Serrin's theorem may be obtained {\it by stability} in the following way: first, for the solution $u$ of the torsion problem prove the estimate $$ r_e-r_i\leq C_t\,\Bigl(\max_{��_t} u-\min_{��_t} u\Bigr) $$ for some constant $C_t$ depending on $t$, where $r_e$ and $r_i$ are the radii of an annulus containing $\partial��$ and $��_t$ is a surface parallel to $\partial��$ at distance $t$ and sufficiently close to $\partial��$; secondly, if in addition $u_��$ is constant on $\partial��$, show that $$ \max_{��_t} u-\min_{��_t} u=o(C_t)\ \mbox{as} \ t\to 0^+. $$ In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains $��$ are ellipses.