  The stability of molecular junctions under transport is of the utmost
importance for the field of molecular electronics. This question is often
addressed within the paradigm of current-induced heating of nuclear degrees of
freedom or current-induced forces acting upon the nuclei. At the same time, an
essential characteristic of the failure of a molecular electronic device is its
changing conductance -- typically from a finite value for the intact device to
zero for a device that lost its functionality. In this publication, we focus on
the current-induced changes in the molecular conductance, which are inherent to
molecular junctions at the limit of mechanical stability. We employ a
numerically exact framework based on the hierarchical equations of motion
approach, which treats both electronic and nuclear degrees of freedom on an
equal footing and does not impose additional assumptions. Studying generic
model systems for molecular junctions with dissociative potentials for a wide
range of parameters spanning the adiabatic and the nonadiabatic regime, we find
that molecular junctions that exhibit a decrease in conductance upon
dissociation are more stable than junctions that are more conducting in their
dissociated state. This represents a new mechanism that stabilizes molecular
junctions under current. Moreover, we identify characteristic signatures in the
current of breaking junctions related to the interplay between changes in the
conductance and the nuclear configuration and show how these are related to
properties of the leads rather than characteristics of the molecule itself.
