  We give a conjecture on the lower bound of the ADM mass $M$ by using the null
energy condition. The conjecture includes a Penrose-like inequality
$3M\geq\kappa\mathcal{A}/(4\pi)+\sqrt{\mathcal{A}/4\pi}$ and the Penrose
inequality $ 2M\geq\sqrt{{\mathcal{A}}/{4\pi}}$ with $\mathcal{A}$ the event
horizon area and $\kappa$ the surface gravity. Both the conjecture in the
static spherically symmetric case and the Penrose inequality for a dynamical
spacetime with spherical symmetry are proved by imposing the null energy
condition. We then generalize the conjecture to a general dynamical spacetime.
Our results raise a new challenge for the famous unsettled question in general
relativity: in what general case can the null energy condition replace other
energy conditions to ensure the Penrose inequality?
