Loose rectangular diagram, multi-crossing number, and arc index

For a non-split multi-crossing diagram $D$ of a link $L$ we show that $α(L)-2 \leq c_2(D) + \sum_{n> 2}(2n-4)c_n(D)$ holds. Here $α(L)$ is the arc index and $c_n(D)$ is the number of $n$-crossings of $D$. This generalizes and subsumes many known inequalities related to multi-crossing numbers. In the course of proof, we introduce a notion of loose rectangular diagram and show that a loose rectangular diagram can be converted to usual rectangular diagram preserving its arc index.