Finite degeneration conjecture to a weak Kähler–Ricci soliton

Let $M$ be a Fano manifold and consider weak conical Kähler–Ricci solitons on varying pairs $(W,\Delta)$, satisfying

$$\operatorname{Ric}(\omega_t)+L_{X_t}\omega_t=t\omega_t+\frac{1-t}{m}[\Delta].$$

A test-configuration for $(M,K_M^{-1})$ is a $\mathbf{C}^*$-equivariant degeneration whose central fiber is a polarized variety. Finite degeneration conjecture. One can solve the displayed equation up to $t=1$ by passing through finitely many singular times and changing the pair $(W,\Delta)$ at each time. At $t=1$ one obtains a $\mathbf{Q}$-Fano variety $W_k$ admitting a weak Kähler–Ricci soliton, and there is a test-configuration for $(M,K_M^{-1})$ with central fiber $W_k$. This conjecture proposes a finite continuation and degeneration process when the original Fano manifold admits no Kähler–Einstein metric. The supplied status evidence says that the cited toric-surface case of the earlier Yau–Tian–Donaldson conjecture is established, but does not resolve this conjecture.