  If $I$ is a monomial ideal with linear quotients, then it has componentwise
linear quotients. However, the converse of this statement is an open question.
In this paper, we provide two classes of ideals for which the converse of this
statement holds. First class is the componentwise polymatroidal ideals in
$K[x,y]$ and the second one is the componentwise polymatroidal ideals with
strong exchange property.
