Correctness preservation for shattered iterations with Borel or Suslin ccc forcing

Let ${\mathbb P}$ be a Borel (or Suslin) ccc forcing, and let $\bar{\mathbb A}$ be a correct diagram. Write

$${\mathbb E}_i={\mathbb A}_i\star{\mathrm{r.o.}}(\dot{\mathbb P}).$$

Correctness-preservation conjecture. The diagram $\bar{\mathbb E}$ is correct.

The preceding lemma proves this preservation for random forcing and its variants, and the authors expect it for a large class of forcing notions. The conjecture remains open in the stated generality.