The vanishing conjecture for free rational modules in locally finite extensions

Let

$$1\longrightarrow T\longrightarrow G\longrightarrow Q\longrightarrow 1$$

be a short exact sequence of countable groups, where $T$ is infinite, locally finite, and abelian, and $Q$ is virtually polycyclic with Hirsch length $h(Q)=d$. A free rational $G$-module is a free module over the group algebra $\mathbb QG$.

Vanishing conjecture. For every free $\mathbb QG$-module $F$,

$$H^i(G,F)=0$$

for all $i\leq d$.

This is presented as a consequence suggested by the uncountable case of the cohomological-dimension conjecture. The source places it among the technical problems and reductions surrounding cohomological dimension for elementary amenable groups; no resolution is supplied in the candidate span.