The p-adic splitting conjecture for definable sets

Let $K$ be a $p$-adically closed field and let $A$ be an infinite definable subset of $K^n$ that is open in its closure. Let $(B_k)_{k\leq q}$ be a finite collection of closed definable subsets of $\overline{A}\setminus A$. The p-adic splitting conjecture. There exists a collection $(A_k)_{k\leq q}$ of non-empty definable subsets of $A$, each clopen in $A$, such that

$$\forall k\leq q,\quad \overline{A_k}=A_k\cup B_k.$$

If true, this would imply that the lattice of definable sets with atoms over $K$ satisfies the ASC-splitting property for every definable set. The source presents the assertion as a conjecture or question, and gives no evidence of resolution.