Deterministic optimal channel proportions for moderate parameters

Let $\alpha,\lambda\in\mathbb{R}_+$, let $f_{\alpha}(\mathbf{a},\mathbf{b},\lambda)$ be the objective function defined by the distributed detection model, and let $\mathbf{a}_\alpha^*$ and $\mathbf{b}_\alpha^*$ be maximizing vectors. Suppose that the second training sequence $Y_2^N$ is pre-processed by a single channel $V\in\mathcal{V}_I$.

Deterministic-maximizer conjecture. For all $\alpha,\lambda\in\mathbb{R}_+$, the vectors $\mathbf{a}_\alpha^*$ and $\mathbf{b}_\alpha^*$ that maximize $f_{\alpha}(\mathbf{a},\mathbf{b},\lambda)$ are deterministic.

Numerical evaluations indicate that, when $\alpha$ is moderate, the maximum occurs at a corner of the feasible set, corresponding to deterministic channel proportions. The claim is not established analytically and remains open.