Suppose $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are two decreasing sequences of positive numbers such that $a_1<b_1$ and $$\sum_{k=1}^\infty a^p_k\leq \sum_{k=1}^\infty b^p_k<\infty$$ for $p=2,3,\cdots.$

Does it follow that $$\sum_{k=1}^\infty a^p_k\leq \sum_{k=1}^\infty b^p_k$$ for all real $p\geq 2$?

$\textbf{Remark 1:}$ Its transparent that the latter inequality cannot be violated by a non-integral sequence $(r_n)$ that runs to infinity. Otherwise by continuity argument $a_1=\max a_n=\displaystyle\lim_{n\to\infty}\|a_n\|_{r_n}\geq\lim_{n\to\infty}\|b_n\|_{r_n} =\max b_n=b_1$!!