Conjecture on the dimension of partial Okounkov bodies

Let $X$ be the compact complex manifold, let $\theta$ be the closed $(1,1)$-form, and let $\theta$-plurisubharmonic function $\varphi\in\operatorname{PSH}(X,\theta)$ be as above, without assuming that $\varphi$ has positive volume. Define the limit partial Okounkov body by

$$\Delta(\theta,\varphi):=\bigcap_{\epsilon\in\mathbb{Q}_{>0}}\Delta(\theta+\epsilon\omega,\varphi).$$

Here $H$ is an ample effective divisor on $X$ and $\omega\in c_1(H)$ is a Kähler form. Dimension conjecture. Under these assumptions,

$$\dim\Delta(\theta,\varphi)=\operatorname{nd}(\theta,\varphi).$$

This conjecture predicts that the dimension of the limit partial Okounkov body is exactly the numerical dimension of the singularity type represented by $\varphi$.