(Semipositivity, convexity, nefness) It is tempting to characterize the semipositivity condition on $\left\lVert\cdot\right\rVert$, in terms of differential conditions on $\Delta_{\mathcal{X}}$, just like convex functions are characterised by the positivity of its Hessian matrix. We speculate that semipositivity should imply that for suitable choices of $h_{\mathcal{L}}$ and $r_{I}$, the curvature form $-dd^{c}\log h_{t}^{1/2}$ can be made positive up to small errors. In the $t\to 0$ limit, the formula (13) then suggests that on each open face $\text{Int}(\Delta_{J})$,

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  The Hessian $D^{2}\phi\geq 0$, namely $\phi$ is convex;

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  The gradient satisfies that $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)\in H^{1,1}(E_{J})$ lies in the nef cone.

Do these two conditions completely characterize semipositive metrics with $\phi=\phi\circ r_{\mathcal{X}}$? If yes, it would naturally explain why (15) defines a measure, instead of just a signed measure.