Linear linking-number bound for polygonal unknots

Let $b3$ be a piecewise linear unknot with vertices $p_i,\dots,p_n$. Let $\varepsilon>0$ be small enough that $N_\varepsilon(b3)$ is a tubular neighborhood. For each choice of points $q_iin B_\varepsilon(p_i)$, let $b2$ be the piecewise linear unknot obtained by connecting $q_1,\dots,q_n$ in order. Linear linking-number conjecture. There is a constant $c$ such that every such $b2$ satisfies

$$\operatorname{lk}(b3,b2)<cn.$$

The proposed estimate would improve the preceding quadratic bound on the Euler number in the circle-bundle construction. The supplied source gives no resolution of this conjecture, so its status is open.