Continuous dependence conjecture for weak solutions of Problem P

Let $T>0$) and $R>0$. Let $\mathcal{X}_{01}=(U_{01},U_{11})$ and $\mathcal{X}_{02}=(U_{02},U_{12})$ belong to $\mathcal{H}_0$, with $\mathcal{X}_{01}$ and $\mathcal{X}_{02}$ having norm at most $R$. Let $\mathcal{X}^1(t)$ and $\mathcal{X}^2(t)$ be weak solutions to Problem P on $[0,T]$ corresponding to these initial data. Continuous dependence conjecture. For every $t\text{ in }[0,T]$, these solutions satisfy

$$\left\|\mathcal{X}^1(t)-\mathcal{X}^2(t)\right\|_{\mathcal{H}_0}\leq e^{Q(R)t}\left\|\mathcal{X}_{01}-\mathcal{X}_{02}\right\|_{\mathcal{H}_0}.$$

Such an estimate would imply continuous dependence of the weak solutions on their initial data and, consequently, uniqueness. The paper identifies proving this estimate as necessary for establishing these properties; its resolution is not supplied here.