Differentiability of the volume function for Kähler classes

Let $X$ be a Kähler manifold of dimension $n$. Let $\alpha$ be a big class and let $\beta$ be a pseudo-effective class. The notation $\langle \alpha^{n-1}\rangle$ denotes the positive intersection product. Differentiability conjecture. One has

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

This proposes a transcendental analogue of the differentiability theorem for volumes of big line bundles. The supplied text gives no resolution in the general Kähler setting.