Unique maximizer of the logarithmic dominant eigenvalue

Let $\mathbf{D}=\operatorname{diag}(d_1,\ldots,d_n)$ with $d_i>0$, and suppose the conditions of Theorem 22 hold. For each $\tau\geq 0$, let

$$U_\tau=\left\{\bar{\mathbf{u}}(\tau)\in\mathbb{R}^n_+: S(\bar{\mathbf{u}}(\tau))\leq\hat S=\gamma^{-1}\ln\frac{K}{\check d}\right\},$$

where $\check d=\min\{d_1,\ldots,d_n\}$. Unique-maximum conjecture. For every $\tau\geq 0$, there exists a unique maximum value over $U_\tau$, namely

$$\ln\bar f(\bar{\mathbf{u}}(\tau))=\gamma S(\bar{\mathbf{u}}),\qquad \bar{\mathbf{u}}\in U_\tau.$$

The claim asserts uniqueness of the maximizing value of the dominant eigenvalue on the convex feasible set; the supplied text does not state whether it has been proved or remains open.