On the linearity defect of the residue field

Given a commutative Noetherian local ring R, the linearity defect of a finitely generated R-module M, denoted ldR(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 ldR(k)<∞ implies ldR(k)=0. We give answers in special cases, and we discuss several interpretations and refinements of the question.