Let $V \subseteq \mathbb{P}^{n_1}_k \times_k \ldots \times_k \mathbb{P}_k^{n_r}$ be a subvariety defined by multihomogeneous polynomials of multidegree at most $(d_1, \ldots, d_r)$, and let $v: \mathbb{P}^{n_1}_k \times_k \ldots \times_k \mathbb{P}_k^{n_r} \rightarrow \mathbb{P}^N_k$ be the Segre embedding.
Is it possible to get a bound on the degree of the image $v(V)$ depending on $d_1, \ldots, d_n$ and $n_1, \ldots, n_r$?
I am really interested in the case $n_1 = \ldots = n_r = 2$, which I do not know if makes things simpler.
I have checked questions in this page such as this but in my case I would like to consider a general subvariety of $\mathbb{P}^{n_1}_k \times_k \ldots \times_k \mathbb{P}_k^{n_r}$ not a product of varieties $V_1 \times_k \ldots \times_k V_r$ with $V_i \subseteq \mathbb{P}_k^{n_i}$.
Thanks in advance.