Lower-bound conjecture for width under connected sums of odd-dimensional manifolds

Let $M_1$ and $M_2$ be closed odd-dimensional triangulated manifolds, and let $M_1 \# M_2$ be any connected sum. Let $\Omega(M)$ denote Thompson's width. Odd-dimensional connected-sum lower-bound conjecture.

$$\max\bigl(\Omega(M_1),\Omega(M_2)\bigr)\leq\Omega(M_1 \# M_2).$$

This addresses the case of odd-dimensional manifolds, which the source says has not otherwise been addressed. The supplied text gives no evidence that the conjecture has been resolved.