Rectifiable Jordan curve with bad polygonal approximations

Question

This question is inspired by this other question of mine. Is there an example of rectifiable Jordan curve $\gamma:[0,1]\to\mathbb R^2$ such that, for some $\delta>0$, no polygonal path $\overline{\gamma(t_0)\gamma(t_1)\cdots\gamma(t_n)}$ with $0=t_0<t_1<\cdots<t_n=1$ and $\max_{1\leq i\leq n}(t_i-t_{i-1})<\delta$, is a Jordan curve? According to this question (with no answer yet), such curve $\gamma$ cannot be piecewise regular.

Answer 1

Lemma 2 in the paper Tverberg answers your question and the paper answers your first post, I think.