Uniqueness conjecture for maximizing Bregman divergence on one side of a codimension-one family

Let $Z$ be the finite sample space, let $u:Z\to\mathbb{R}$ satisfy $\sum_{x\in Z}u(x)=0$, and let $\mathcal{F}_{u}$ be the exponential family associated with a function $f_u$ such that $\mathcal{N}(f_u)=\mathbb{R}u$. Define

$$\mathcal{P}_{u}^{+}=\big\{P\in\mathcal{P}(Z):\langle P-\Pi_{\mathcal{F}_{u},P},u\rangle>0\big\}.$$

Uniqueness conjecture. The map

$$P\in\mathcal{P}_{u}^{+}\longmapsto B(P,\mathcal{F}_{u})$$

has a unique local and global maximizer. For a codimension-one exponential family, this predicts uniqueness of the maximizing distribution on each side of the family, complementing the stated fact that there are precisely two local maximizers overall, one on each side. Whether this uniqueness holds in the Bregman-divergence setting is left open by the source.