Tensor product/Flags/Exercise

Let ${}V_{1},\ldots ,V_{n}$ be finite-dimensional ${}K$-vector spaces, and let

$U_{1,1}\subset U_{1,2}\subset \ldots \subset U_{1,\dim _{K}{\left(V_{1}\right)}}\subset V_{1},\ldots ,U_{n,1}\subset U_{n,2}\subset \ldots \subset U_{n,\dim _{K}{\left(V_{n}\right)}}\subset V_{n}$

be flags in these vector spaces. Show that, in general, there does not exist a flag in ${}V_{1}\otimes _{K}\cdots \otimes _{K}V_{n}$ such that the linear subspaces involved have the form

$U_{1,j_{1}}\otimes _{}\cdots \otimes _{}U_{n,j_{n}}.$

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