Geography of log adjunction

Let $X/Z$ be a proper morphism, let $\eta$ be the general point of $Z$, and let $S_i$ be distinct prime b-divisors of $X$. Let $\mathfrak P$ be a polyhedron in the wlc geography of $\mathfrak N_{S_\eta}$ of $(X_\eta/\eta,\sum S_{i,\eta})$. Let $D$ be a b-divisor and let $\mathcal D{}^{\mathrm{mm}}(D)$ denote the corresponding maximal moduli b-divisor. Geography of log adjunction. The function $\mathcal D{}^{\mathrm{mm}}(D)$ is linear on $\overline{\mathfrak P}$: for every two b-divisors $D_1,D_2$ such that $D_{1,\eta},D_{2,\eta}\in\overline{\mathfrak P}$ and any $w_1,w_2\in[0,1]$ with $w_1+w_2=1$,

$$\mathcal D{}^{\mathrm{mm}}(w_1D_1+w_2D_2)=w_1\mathcal D{}^{\mathrm{mm}}(D_1)+w_2\mathcal D{}^{\mathrm{mm}}(D_2).$$

Equivalently, $\mathcal D{}^{\mathrm{mm}}$ is piecewise linear as a function of the horizontal part $D_\eta$ of $D$. This is a stronger stability assertion for log adjunction and is stated after the b-semiampleness conjecture; the supplied text gives no resolution.