Boundedness of finite subgroups of rational Cremona group of rank 3

In 2009 J.-P. Serre providied an explicit multiplicative bound for orders of finite subgroups of plane Cremona group of rank 2 over finitely generated fields over $Q$ and asked a question: is it true that order of finite subgroups of Cremona groups of arbitrary rank over mentioned fields are bounded and how the bound can be estimated?
In 2013, Yu. Prokhorov and C. Shramov answered the first part of the question in more general setup. They proved that group of birational automorphisms of arbitrary variety of arbitrary dimension over finitely generated fields over $Q$ has bounded finite subgroups. But their bounds were no longer explicit. I will discuss how we can try to answer the second part of Serre’s question in case of $Cr_3(Q)$.