  The most important open problem in Monotone Operator Theory concerns the
maximal monotonicity of the sum of two maximal monotone operators provided that
Rockafellar's constraint qualification holds.
  In this paper, we prove the maximal monotonicity of $A+B$ provided that $A$
and $B$ are maximal monotone operators such that $\dom A\cap\inte\dom
B\neq\varnothing$, $A+N_{\overline{\dom B}}$ is of type (FPV), and $\dom
A\cap\overline{\dom B}\subseteq\dom B$. The proof utilizes the Fitzpatrick
function in an essential way.
