Regularity and $v$-number of syzygies of graded maximal Cohen-Macaulay modules

Let $A=Q\mathrm{/}(\mathbf{f})$, where $Q$ is a polynomial ring over an infinite field $k$ and $\mathbf{f}=f_1,\dots ,f_c$ is a homogeneous $Q$-regular sequence. For $1\le j\le c$, set $w_j=\mathrm{deg}(f_j)>0.$ Let $M$ be a finitely generated graded $A$-module of infinite projective dimension. Set $M_i={\mathrm{Syz}}_i^A(M)$. We prove that, for each $l\in \{0,1\}$, the sets ${\mathrm{Ass}}_A(M_{2i+l})$ eventually stabilize, and that $\mathrm{reg}(M_{2i+l})$ is eventually bounded above and below by linear polynomials. If $w_j=w$ for all $1\le j\le c$, then the functions $i⟼\mathrm{reg}(M_{2i+l})$, $i⟼v(M_{2i+l}),$ and $i⟼v_{\mathfrak{p}}(M_{2i+l}),\mathfrak{p}\in {\mathcal{A}}_l,$ are eventually linear with leading coefficient $w$, where ${\mathcal{A}}_l$ denotes the stable set of associated primes.