The power conjecture for componentwise polymatroidal ideals

Let $I\subset S$ be a componentwise polymatroidal ideal in a polynomial ring $S=K[x_1,\dots,x_n]$. Power conjecture. Every power $I^k$ of $I$ has linear quotients. The source gives this as an expectation after noting that products of componentwise polymatroidal ideals need not remain componentwise polymatroidal; no resolution is supplied in the provided text.