  Given an immersed, Maslov-$0$, exact Lagrangian filling of a Legendrian knot,
if the filling has a vanishing index and action double point, then through
Lagrangian surgery it is possible to obtain a new immersed, Maslov-$0$, exact
Lagrangian filling with one less double point and with genus increased by one.
We show that it is not always possible to reverse the Lagrangian surgery: not
every immersed, Maslov-$0$, exact Lagrangian filling with genus $g \geq 1$ and
$p$ double points can be obtained from such a Lagrangian surgery on a filling
of genus $g-1$ with $p+1$ double points. To show this, we establish the
connection between the existence of an immersed, Maslov-$0$, exact Lagrangian
filling of a Legendrian $\Lambda$ that has $p$ double points with action $0$
and the existence of an embedded, Maslov-$0$, exact Lagrangian cobordism from
$p$ copies of a Hopf link to $\Lambda$. We then prove that a count of
augmentations provides an obstruction to the existence of embedded, Maslov-$0$,
exact Lagrangian cobordisms between Legendrian links.
