Bogatyi–Shavgulidze covering-system periodic point conjecture

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous, let $I_1,\ldots,I_k$ be closed intervals, and call $(I_1,\ldots,I_k;f)$ a covering system when

$$f(I_1\cup\cdots\cup I_k)\supseteq I_1\cup\cdots\cup I_k.$$

Bogatyi–Shavgulidze conjecture. For every covering system $(I_1,\ldots,I_k;f)$, there exists $x_0\in I_1\cup\cdots\cup I_k$ and an integer $l\leq k$ such that

$$f^l(x_0)=x_0.$$

The conjecture is known for $k\leq 5$; the general case remains open.