Existence conjecture for $(1,1,2\pi[D])$-extremal Poincaré-type Kähler metrics

Let $(X,\omega)$ be a compact Kähler manifold and let $D$ be a smooth divisor. Let $T$ be a maximal compact torus in the reduced automorphism group $\operatorname{Aut}_{\mathrm{red}}(X,D)$. Write $\ell^{\mathrm{ext}}$ for the twisted extremal function associated to the twist $2\pi[D]$, and $\ell^{\mathrm{ext}}_D$ for the extremal function intrinsic to $(D,[\omega]|_D,T)$. A $T$-invariant $(1,1,2\pi[D])$-extremal metric is a Poincaré-type Kähler metric whose potential satisfies the corresponding twisted extremal equation. Existence conjecture for $(1,1,2\pi[D])$-extremal Poincaré-type Kähler metrics. The manifold $X$ admits such a metric in $[\omega]$ if and only if: (i) the twisted K-energy $\mathcal{M}_{1,\ell^{\mathrm{ext}}}^{2\pi[D]}$ is coercive on $\mathcal{E}^{1,T}(X,\omega)$ relative to $T^\mathbb{C}$; (ii) the extremal K-energy $\mathcal{M}_{1,\ell^{\mathrm{ext}}_D}$ is coercive on $\mathcal{E}^{1,T}(D,\omega|_D)$ relative to $T^\mathbb{C}$; and (iii) on $D$,

$$\ell^{\mathrm{ext}}_D-\ell^{\mathrm{ext}}|_D=c>0$$

for some constant $c$. The conjecture proposes an analytic characterization of these extremal Poincaré-type metrics through global coercivity, boundary coercivity, and a boundary stability condition. It is motivated by the relationship between Poincaré-type extremal metrics and relative K-stability, as well as by known necessary conditions and toric results, but the supplied source does not establish the claimed equivalence.