Global-minimum and strict-saddle conjecture for the third-layer synaptic objective

Let $X \in \mathbb{R}^{n \times T}$ satisfy $C_X \succ 0$, and let $W(t)$ follow the continuous-time gradient-flow feedforward synaptic dynamics. Let $\mathcal{S}$ denote the stationary-point set of the third-layer objective $S_3$. If $\Lambda_C^m=\operatorname{diag}(\lambda_1^C,\dots,\lambda_m^C)$ contains the $m$ largest eigenvalues of $C_X$, and $V_C^m=(v_1^C,\dots,v_m^C)$ contains the corresponding eigenvectors, then global-minimum and strict-saddle conjecture. The global minima of $S_3(W)$ are precisely the points $W^\star\in\mathcal{S}$ of the form

$$W^\star=U\Lambda_C^m(V_C^m)^\top,\qquad U\in\mathcal{O}_m.$$

Every other $\bar W\in\mathcal{S}$ is a strict saddle or a maximum. This conjecture concerns the landscape of the third-level optimization problem and is supported by the paper's heuristic analysis and empirical validation, but no proof or resolution is supplied.