Maximal degree of elements of minimal generating set

A) Let an integer sequense $s_i$ has a reccurence relation $s_{n+d}+a_1 s_{n+d-1}+\cdots +a_d s_n=0$ and let a generating functions of the sequence is $\frac{P(t)}{Q(t)}=\sum_i s_i t^i,$ where $P(t),Q(t)$ are coprime polynomials, $Q(t)=1+a_1t+a_2 t^2+\cdots+a_d t^d.$

B) Let $C$ is Cohen-Macalаy algebra, let $f_1,f_2,\ldots,f_m$ is its minimal generating set, let $P(t)/Q(t)$ (as above) is its Poincare series and let $\deg Q(t)=\deg P(t)+\rm{tr deg}C+1.$

Question. Does the conditions A) ,B) implies that $\max \deg(f_i) \leq \deg Q(t)=d?.$

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