Linearity of the logarithmic dominant eigenvalue on the feasible set

Let $\mathbf{D}=\operatorname{diag}(d_1,\ldots,d_n)$ with $d_i>0$, and suppose the conditions of Theorem 22 hold: $\lambda^*(\mathbf{D}^{-1}\mathbf{Q}_m)>1$ for every $\mathbf{M}\in\mathbf{M}_K(\tau)$ and $\mathbf{Q}(\tau)\in R(\tau)$, for all $\tau\geq 0$. 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 $\check d=\min\{d_1,\ldots,d_n\}$. Linearity conjecture. For each $\tau\geq 0$, the function $\ln\lambda(\bar{\mathbf{u}}(\tau))=\ln\bar f(\bar{\mathbf{u}}(\tau))$ is a linear functional on $U_\tau$. Here $\lambda(\bar{\mathbf{u}}(\tau))=\bar f(\bar{\mathbf{u}}(\tau))$ is the dominant eigenvalue of the problem. The claim concerns the dependence of the dominant eigenvalue on the state over the convex feasible set; the supplied text gives no resolution or further context establishing whether it is proved.