Burton's conjecture on intersections of normal-surface-theory fans

For $d\geq2$, let $U(d,k)$ denote the maximum number of inclusion-maximal cones in the intersection of $k$ balanced fans in general position in $\mathbb{R}^d$, where each fan consists of three halfplane cones with a common axis. Burton's conjecture. The maximum for $k=d-1$ satisfies

$$U(d,d-1)=1+2^{d-1}.$$

The source states that its Minkowski-sum results resolve this conjecture, so the claim is solved.