In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good groups.

In Champs Affines (ArXiv), Toën introduces an analogous definition of an *algebraically good* group (Definition 3.4.1, p 84). In particular, a discrete group $G$ is algebraically good if and only if the pro-algebraic completion morphism $G\to G^{\mathrm{alg}}$ induces an isomorphism
$$
H^{i}(G^{\mathrm{alg}},M)\to H^{i}(G,M)
$$
for every (finite-dimensional) $G$-representation $M$.

My first question is simple: Are the braid groups algebraically good? What about the pure braid groups?

A more interesting question is the following: Under what constructions is algebraic goodness preserved?

For instance, when is an extension of algebraically good groups good?