The gradient-flow global-maximization conjecture for image-labeling assignments

Let $\mathcal{W}$ be the space of admissible assignment matrices, let $W(t)$ be the flow generated by the specified gradient-flow equation, and let $\overline{\mathcal{W}}^{\ast}$ denote the set of global maxima of the objective over the relevant closure. For any fixed data and prior data, except on a subset of $\mathcal{W}$ of measure zero, the flow approaches this set in the following sense: for every $0<\varepsilon\ll 1$, there exists $t=t(\varepsilon)$ and $\overline{W}^{\ast}\inoverline{\mathcal{W}}^{\ast}$ such that

$$\left\|W\bigl(t(\varepsilon)\bigr)-\overline{W}^{\ast}\right\|\leq\varepsilon.$$

Gradient-flow global-maximization conjecture. For any data and prior data, up to a subset of $\mathcal{W}$ of measure zero, the flow $W(t)$ approximates a global maximum in the stated sense.