Let $b_1,b_2,\dots$ be positive real numbers such that $$s_1<\infty\quad\text{and}\quad z_1<\infty, $$ where $$s_k:=\sum_{j=k}^\infty b_j\quad\text{and}\quad z_k:=\sum_{j=k}^\infty\frac{b_j}{\sqrt{s_j}} $$ for natural $k$. Does it then necessarily follow that $$\limsup_{k\to\infty}\frac{z_k}{\sqrt{s_k}}<\infty\,\text{?} $$ (One may note here that $z_k\ge\sqrt{s_k}$ for all $k$, whence $\liminf_{k\to\infty}\frac{z_k}{\sqrt{s_k}}\ge1$.)

(This question is a "discretized" version of a problem considered in this answer.)