The recursive demipotent conjecture for meet semi-lattices

Let $L$ be a meet semi-lattice with a fixed linear extension, and let $D$ range over the valid diagrams of $L$, meaning subsets containing the minimal elements and stable under joins. For each valid diagram $D$, let $C_D=L_DR_D$ be the element constructed recursively from the restriction of $L$ and $D$ as in the branching construction. Recursive demipotent conjecture. The set

$$\{C_D\mid D\text{ valid diagram}\}$$

forms a set of demipotent elements for $\mathcal{OR}(L)$ which, after raising each element to a sufficiently high power, yields a set of primitive orthogonal idempotents. The claim would extend the branching decomposition from the discussed examples to arbitrary meet semi-lattices and provide primitive idempotents through powers of the recursively constructed elements. The supplied text presents it as conjectural and gives no resolution.