Conjecture on the exceptional three-part Cohen–Macaulay arrangements

Let $b>c$ and let $q,r,s$ be positive integers. For the partition

$$\lambda=((b+c)^q,b^r,c^s),$$

let $X_\lambda$ denote the associated subspace arrangement. Three-part non-Cohen–Macaulayness conjecture. If $q>1$, then $X_\lambda$ is not Cohen–Macaulay; consequently, $X_\lambda$ is not Cohen–Macaulay for every partition having at least three distinct parts. The conjecture is supported by computational evidence for $q=2$ and $r=s=1$, but remains open in the stated generality.