Matrix factorization with non-negative entries

Matrix factorization with non-negative entries

Suppose $H\in \mathbb{R_+}^{m\times n}$ is a matrix of rank
$r,~2<r<\max(m,n)$ with non-negative entries. Can we factorize $H$ as
$H=VW$ such that $V\in \mathbb{R_+}^{m\times r}$ and $W\in
\mathbb{R_+}^{r\times n}$ where both $V,W$ both have non-negative entries?
Note that if $V,W$ are allowed to have arbitrary entries rather than
non-negative entries, the answer is in the affirmative. The columns of $H$
span a subspace of dimension $r$. Let $V\in \mathbb{R}^{m\times r}$ be a
matrix whose columns form a basis for this subspace. Then $H=VW$ where
column $i$ of $W$ are the coefficients of column $i$ of $H$ with respect
to the basis vectors in $V$. The difficulty arises due to the
non-negativity constraint.
(Note: I have a proof for ranks 1,2 and full rank.)