A note on linear resolution and polymatroidal ideals

Let $R=K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: $(i)$ ${\rm height}(I)=n-1$;$(ii)$ $I$ contains at least $n-3$ pure powers of the variables $x_1^d,\mathrm{.}\mathrm{.}\mathrm{.},x_{n-3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables.