The eventual shape of the Betti table of $\mathfrak{m}^kM$

Let $S$ be the polynomial ring over a field $K$ in a finite set of variables, and let $\mathfrak{m}$ be the graded maximal ideal of $S$. It is known that for a finitely generated graded $S$-module $M$ and all integers $k\gg 0$, the module $\mathfrak{m}^kM$ is componentwise linear. For large $k$ we describe the pattern of the Betti table of $\mathfrak{m}^kM$ when $\mathrm{char}(K)=0$ and $M$ is a submodule of a finitely generated graded free $S$-module. Moreover, we show that for any $k\gg 0$, $\mathfrak{m}^kI$ has linear quotients if $I$ is a monomial ideal.