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It is known that if $\\mathcal{C}$ is chordal, then $I(\\bar{\\mathcal{C}})$ has a linear resolution over all fields. The converse has recently been rejected, but the following question which poses a weaker version of the converse is still open: \"if $I(\\bar{\\mathcal{C}})$ has linear quotients, is $\\mathcal{C}$ necessarily chordal?\". 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