This is precisely what happens in bubbling. Wwe've already ruled out the case that $u^i(z_i)$ runs into the "infinitely stretched neck" (as $i\to\infty$), that's Case#1 in the paper. So the only concentration of energy is some place near the "bulk" $\Sigma'$ which then forms a bubble. Note that the semi-infinite ends attached to $\Sigma'$ are just the result of stretching the neck, but by assumption our points $u^i(z_i)$ don't run away over there.