Conjecture on groups of product type or K3 type for rationally connected threefolds

Let $X$ be a rationally connected variety of dimension $3$, and let $G\subset \mathrm{Bir}(X)$ be a finite abelian group. By the third type, we mean the case in which $G$ faithfully acts on a $G\mathbb{Q}$-Fano threefold $X$ with $|-K_X|=\emptyset$, as in Theorem 1. Product-or-K3 type conjecture. Any group of the third type is either of product type or of K3 type. Here a group is of product type if $G\simeq G_1\times G_2$ with $G_i\subset \mathrm{Cr}_i(\mathbb{C})$, while it is of K3 type if it is an extension of a finite abelian group faithfully acting on a K3 surface by a cyclic group. The conjecture would eliminate the third case as a genuinely new possibility in the classification of finite abelian groups acting on rationally connected threefolds; the source presents it as an expectation, and no resolution is supplied here.