The invariant-ring sum conjecture for infinite-type Cartan matrices

Let $A$ be an $n\times n$ Cartan matrix of infinite type. Let $P$ be the ring of invariants associated with the Weyl group of $A$, and let $P_i$ be the corresponding invariant subrings for the rank-one subsystems indexed by $i$. Invariant-ring sum conjecture.

$$P=P_1+P_2+\cdots+P_n.$$

This conjecture is motivated by Foley's claim for $3\times3$ Cartan matrices of infinite type. The general equality is presented as an open conjecture.