  Recently, advances in film synthesis methods have enabled a study of
extremely overdoped $La_{2-x}Sr_{x}CuO_{4}$. This has revealed a surprising
behavior of the superfluid density as a function of doping and temperature, the
explanation of which is vividly debated. One popular class of models posits
electronic phase separation, where the superconducting phase fraction decreases
with doping, while some competing phase (e.g. ferromagnetic) progressively
takes over. A problem with this scenario is that all the way up to the dome
edge the superconducting transition remains sharp, according to mutual
inductance measurements. However, the physically relevant scale is the Pearl
penetration depth, $\Lambda_{P}$, and this technique probes the sample on a
length scale $L$ that is much larger than $\Lambda_{P}$. In the present paper,
we use local scanning SQUID measurements that probe the susceptibility of the
sample on the scale $L << \Lambda_{P}$. Our SQUID maps show uniform landscapes
of susceptibility and excellent overall agreement of the local penetration
depth data with the bulk measurements. These results contribute an important
piece to the puzzle of how high-temperature superconductivity vanishes on the
overdoped side of the cuprates phase diagram.
