Flat-metric convergence conjecture for degenerating conformal metrics on the 2-sphere

Let $M$ be a topological $2$-sphere, and let $g_i=e^{2u_i}g_0$ be a family of conformal metrics with bounded area and energy. Assume

$$\lim_{i\rightarrow\infty}\int_M |\nabla K_{g_i}|^2\,d g_0=0.$$

Suppose there is exactly one bubble point $p\in S^2$ and $u_i\rightarrow-\infty$ on every compact subset $\Omega\subset M\setminus\{p\}$. For a fixed point $q\in\Omega$, set $c_i=u_i(q)$. Flat-metric convergence conjecture. There exists a flat metric $g_\infty=e^{2u_\infty}g_0$ on $M\setminus\{p\}$ and a subsequence of $g_i$ such that, for every compact subset $\Omega\subset M\setminus\{p\}$,

$$u_i-c_i\rightharpoonup u_\infty\quad\text{in }H^{2,2}(\Omega).$$

The conjecture describes the limiting geometry away from the unique concentration point when the curvature-gradient energy tends to zero. Its status is not determined by the supplied source context.