Let $V$ and $W$ be finite-dimensional vector spaces over a field $k$ with (exhaustive, separated, finite, descending) filtrations $F^\bullet$ and $G^\bullet$, respectively. On $V \otimes_k W$, we can form the tensor filtration $Fil^\bullet$ associated to $F^\bullet$ and $G^\bullet$ as follows: $$Fil^k(V \otimes W) = \sum_{i+j = k} F^i V \otimes G^j W$$ To any filtered vector space we can associate a Hodge polygon as follows. Consider the filtered vector space $(V, F^\bullet)$. We define the associated grading by $$gr_F^i (V) = F^i V / F^{i+1} V.$$ Let $\{i_0 < \dots < i_n\}$ be the distinct indices such that $gr_F^i(V) \neq 0$. Then the Hodge polygon $H(V, F^\bullet)$ is the convex polygon in $\mathbb R^2$ starting at $(0, 0)$ with $\dim_k gr_F^{i_j}(V)$ segments of horizontal distance $1$ and slope $i_j$ for $0 \leq j \leq n$.
My question: is there a geometric relationship between the Hodge polygons $H(V, F^\bullet)$ and $H(W, G^\bullet)$ on the one hand and $H(V \otimes W, Fil^\bullet)$ on the other?
I'd be happy to make the simplifying assumption that there exist $k< l$ such that $V = F^k V \neq F^{k+1} V$ and $W = G^k W \neq G^{k+1} W$ and $F^l V \neq F^{l+1} V = 0$ and $G^l W \neq G^{l+1} W = 0$, i.e. that both filtrations "start" and "end" at the same time.
Having computed some examples, I haven't been able to find a clear relationship except in very simple cases.