Differentiability and orthogonality conjectures for positive intersection products

Let $X$ be a compact Kähler manifold, let $n=\dim X$, and let $\alpha$ be a big class. For a cohomology class $\gamma\in H^{1,1}(X,\mathbb{R})$, write $\langle\alpha^k\rangle$ for the positive intersection product, and write $\alpha=\langle\alpha\rangle+N(\alpha)$ for the divisorial Zariski decomposition of $\alpha$.

Differentiability and orthogonality conjectures. The following two assertions are conjectured:

1. For every $\gamma\in H^{1,1}(X,\mathbb{R})$,

$$\left.\frac{d}{dt}\right|_{t=0}\langle(\alpha+t\gamma)^n\rangle=n\gamma\cdot\langle\alpha^{n-1}\rangle.$$

2. The orthogonality relation

$$\langle\alpha^{n-1}\rangle\cdot N(\alpha)=0$$

holds.

These conjectures concern the differentiability of the volume or positive intersection product and the orthogonality of the positive and negative parts in the divisorial Zariski decomposition of a big cohomology class. The source relates them to the assumption that pullbacks of big classes have zero intersection with exceptional divisors; their resolution status is not specified here.