Additivity of the log Chern character for distinguished triangles

Let $(X,D)$ be a log smooth pair, and let $E_1\to E_2\to E_3\to E_1[1]$ be a distinguished triangle in $D(X\times^{\log}X)$ whose terms are strong log Fourier–Mukai kernels. Additivity conjecture. The log Chern character satisfies

$${\rm ch}^{\log}(E_1)+{\rm ch}^{\log}(E_3)={\rm ch}^{\log}(E_2).$$

If true, this would extend the already established additivity for direct sums and make the log Chern character define a morphism out of the $K$-theory $K^0(X)$ of perfect complexes.