The infinite-generation conjecture for rational cohomology of indefinite Kac-Moody groups

Let $A=(a_{ij})_{n\times n}$ be a Cartan matrix of indefinite type with $n\geq3$, and let $G(A)$ be the associated Kac-Moody group. Infinite-generation conjecture. The rational cohomology ring

$$H^*(BG(A);\mathbb Q)$$

is infinitely generated.

The conjecture is motivated by the authors' result that the rational cohomology ring is infinitely generated for a generic Kac-Moody group. The assertion for every indefinite-type Cartan matrix of rank at least $3$ is presented as open.