Birkhoff–von Neumann like linear algebra proof

Question

I have an $m \times n$ matrix with entries in $[0,1]$ which is constrained by some equality (and some inequality) constraints. The equality constraints are of the form:

$x_{i_1,j_1}+\cdots+x_{i_k,j_k}=x_{l_1,m_1}+\cdots+x_{l_k,m_k}$ for all $k \in \{2,\cdots,n\}$, for some given $n$. These sums don't have be some specific number (unlike in the Birkhoff polytope, which also involves such equality constraints, but they sum to $1$), but they have to be $\leq 1$.

The inequality constraints are of the form:

$x_{i'_1,j'_1}+\cdots+x_{i'_N,j'_N} \leq 1$

Hence the set of all such matrices is a polytope (more specifically, a bounded polyhedral cone), say $\mathcal{B}$. I want to prove that such a matrix is necessarily a convex combination of matrices satisfying the same constraints, but with entries in $\{0,1\}$, a la the Birkhoff–von Neumann theorem, as proved here.

I'm trying to use the same strategy as the above proof, i.e. showing that the extreme points of $\mathcal{B}$ are integral, i.e. any matrix in $\mathcal{B}$ with an element in $(0,1)$ cannot be an extreme point. However I'm facing the following two issues (as of now):

1. The aforementioned proof is based on forming a cycle of $x_{i,j}$'s which are all in $(0,1)$. Then we alternately add and subtract a suitable $\epsilon$ from each $x_{i,j}$ in the cycle to form a different matrix within the Birkhoff polytope (which shows our starting point was not an extreme point). This takes care of all equality constraints since each $x_{i,j}$ is part of exactly two equality constraints. However in my case each $x_{i,j}$ is part of a multitude of equality constraints. The number of such constraints which each $x_{ij}$ is part of is also not the same. Hence I'm not able to determine how I should go about forming such a cycle in this case. Even if we do form one, and do the $\epsilon$-adjustments to reach a new point in $\mathcal{B}$, there may be other equality constraints which cease to hold because of these $\epsilon$-adjustments, because each element is part of many such constraints.

2. The inequality constraints could hold with equality even if all terms are strictly in $(0,1)$. This was not the case for the Birkhoff polytope. If some of the inequality constraints hold with equality then these give rise to a different set of equality constraints in this case.

Any help is deeply appreciated. Thank you.