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The most important open problem in Monotone Operator Theory concerns themaximal monotonicity of the sum of two maximally monotone operators providedthat Rockafellar's constraint qualification holds.  In this paper, we prove the maximal monotonicity of $A+\partial f$ providedthat $A$ is a maximally monotone linear relation, and $f$ is a proper lowersemicontinuous convex function satisfying $\dom A\cap\inte\dom \partialf\neq\varnothing$. Moreover, $A+\partial f$ is of type (FPV). The maximalmonotonicity of $A+\partial f$ when $\intdom A\cap\dom \partialf\neq\varnothing$ follows from a result by Verona and Verona, which the presentwork complements.

The Sum of a Maximally Monotone Linear Relation and the Subdifferential of a Proper Lower Semicontinuous Convex Function is Maximally Monotone