Let $R=K[x_1,...,x_n]$ 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(BH-Conjecture) that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have linear resolution. Herzog, Hibi and Zheng proved that if $I$ is a monomial ideal generated in degree $2$, then $I$ has a linear resolution if and only if each power of $I$ has a linear resolution. Sturmfels gave an example $I=(def,cef,cdf,cde,bef,bcd,acf,ade)$ with $I$ has a linear resolution while $I^2$ has no linear resolution. This suggests the following question: Is it true that each power of $I$ has a linear resolution, if $I$ is a squarefree monomial ideal of degree $d$ with $I^k$ has a linear resolution for all $1\leq k\leq d-1$? In this talk we speak about BH-Conjecture and we give an affirmative answer in the following cases: $(i)$ $\height(I)=n-1$; $(ii)$ $I$ contains at least $n-3$ pure powers of the variables $x_1^d,...,x_{n-3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables
