Unique-largest-antichain conjecture for balls in the Boolean lattice

Let $[n]=\{1,\dots,p\}\cup\{p+1,\dots,p+q\}$ with $q=n-p$. For $i,j\geq 0$, let $X_{i,j}$ be the family of sets obtained from $[p]$ by subtracting $i$ elements and adding $j$ elements from $\{p+1,\dots,p+q\}$. Define the ball of radius $r$ centered at $[p]$ by

$$B_r[p,q]=\bigcup_{i+j\leq r}X_{i,j}.$$

Unique-largest-antichain conjecture. The largest layer of $B_r[p,q]$ is its unique largest antichain.

This concerns the width of a structured subposet of the Boolean lattice. The surrounding discussion presents determining or bounding the width of such subposets as a wide-open problem; the supplied text gives no resolution of this claim.