  Given a commutative Noetherian local ring $R$, the linearity defect of a
finitely generated $R$-module $M$, denoted $\ld_R(M)$, is an invariant that
measures how far $M$ and its syzygies are from having a linear resolution.
Motivated by a positive known answer in the graded case, we study the question
of whether $\ld_R(k)<\infty$ implies $\ld_R(k)=0$. We give answers in special
cases, and we discuss several interpretations and refinements of the question.
