An isoperimetric inequality for “order” polytopes

I am looking for an isoperimetric inequality for order-like polytopes. An order polytope $K\in \mathbb{R}^n$ is defined by a set of linear inequaities: $$\forall i \; 0\leq x_i \leq 1$$ and $x_i \leq x_j$ for some of the coordinates. If the constraints define a total order over the coordinates, the polytope $K$ is called a total order polytope.

As in a related question, given a parameter $k$, I am interested in how large can the ratio between the number of boxes which intersect the perimeter of $K$ and the number of boxes contained inside the polytope.

One thought is to think of this as an isoperimetric inequality - if we would expand the polytope by a constant factor, and we knew how much the volume would change - I think that it might help us get that ratio.

Thanks!

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