Connected-sum superadditivity conjecture for Thompson's width

Let $M_1$ and $M_2$ be $n$-dimensional simplicial manifolds, and choose a connected sum $M_1 \# M_2$. Here $\Omega(M)$ denotes Thompson's width of a simplicial manifold $M$. Connected-sum superadditivity conjecture. There exist $M_1$, $M_2$, and a choice of connected sum such that

$$\Omega(M_1 \# M_2)>\Omega(M_1)\cup\Omega(M_2).$$

The conjecture asks whether the upper bound obtained from particular choices of bricks can fail for a suitable connected sum. It is presented as open in the source; related work showed that Gabai's knot width is not additive under connected sum, while a modified width can be additive for certain 3-manifold–graph pairs.