On the polymatroidality of monomial ideals

Let $S=K[x_1,ldots,x_n]$ be the polynomial ring over a field $K$ and let $Isubset S$ be a monomial ideal. We say that $I$ has quotients with linear resolution with respect to the ordering $u_1,ldots,u_r$ of minimal generators whenever for all $j$, the colon ideal $(u_1,ldots,u_{j-1}):u_j$ and $I$ itself have a linear resolution. The aim of this paper is to discuss the following question: if $I$ has quotients with linear resolution with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables then can we say that $I$ is polymatroidal?