Mabuchi–Nakagawa conjecture for canonical cones of Kähler–Ricci solitons

Let $X$ be a smooth Fano manifold admitting a Kähler–Ricci soliton, and let $Y:=K_X^{\times}$ be the complement of the zero section in its canonical bundle. A Calabi–Yau cone structure means a Ricci-flat Kähler cone structure on $Y$. Mabuchi–Nakagawa conjecture. The canonical cone $Y:=K_X^{\times}$ admits a Calabi–Yau cone structure. The paper explains that this conjecture is a special case of the problem studied there, while noting that it generally fails for Fano orbifolds.