Existence of good minimal models for klt adjoint foliated structures

Let $(X,\mathcal{F},t)$ be a projective klt adjoint foliated structure, and set

$$K:=tK_{\mathcal{F}}+(1-t)K_X.$$

Assume that $K$ is pseudo-effective and $t<1$. Existence of good minimal models. The structure $(X,\mathcal{F},t)$ has a good minimal model: there exists a $K$-negative birational contraction $\phi\colon X\dashrightarrow X_{\min}$ such that $\phi_*K$ is semi-ample. If $K$ is big, then $(X,\mathcal{F},t)$ has a canonical model: there exists a $K$-non-positive birational contraction $\psi\colon X\dashrightarrow X_{\operatorname{can}}$ such that $\psi_*K$ is ample. Moreover, $X_{\operatorname{can}}$ is unique, and if $t$ is rational, then

$$X_{\operatorname{can}}=\operatorname{Proj}\bigoplus_{m=0}^{+\infty}H^0\left(X,\mathcal{O}_X\left(\left\lfloor m\left(tK_{\mathcal{F}}+(1-t)K_X\right)\right\rfloor\right)\right).$$

This is a conjectural BCHM-type statement for adjoint foliated structures, strengthening the theorem discussed immediately beforehand. It predicts existence of good minimal models in the pseudo-effective case and canonical models in the big case, including uniqueness and the stated Proj description for rational $t$.