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Let $(P,\le)$ be an abstract $n$-polytope, with $n\ge 2$. Let $H,H',K$ be $m$-faces, with $0\le m \le n-2$. Is it true that there is a sequence $\{H_0=H,H_1,\ldots,H_{r-1},H_r=H' \} \subseteq P$ so that $H_i$ is incident with $H_{i+1}$ for each $i$, $H_i$ is an $m$-face when $i$ is even, and $H_i$ is an $(n-1)$-face so that $K \not \le H_i$ when $i$ is odd? I know it's a strange question, and it follows quite easily from connectedness that, without the last requirement ($K \not \le H_i$) it is true.

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