Linearity defect of the residue field of short local rings

Let $(R,\m,k)$ be a Noetherian local ring with maximal ideal $\m$ and residue field $k$. The linearity defect of a finitely generated $R$-module $M$, which is denoted $\ld_R(M)$, is a numerical measure of how far $M$ is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether $\ld_R(k)<\infty$ implies $\ld_R(k)=0$, in the case when $\m^4=0$.