Koszul binomial edge ideals of pairs of graphs

We study the Koszul property of a standard graded $K$-algebra $R$ defined by the binomial edge ideal of a pair of graphs $(G_1,G_2)$. We show that the following statements are equivalent: (i) $R$ is Koszul; (ii) the defining ideal $J_{G_1,G_2}$ of $R$ has a quadratic Gr"obner basis; (iii) the graded maximal ideal of $R$ has linear quotients with respect to a suitable order of its generators