Didid's zone-width and inradius conjecture

Let $F4D0\mathbf{Z}_1,\dots,\mathbf{Z}_n\subset \mathbb{S}^{d-1}$, with $d>2$, be zones of widths $2\beta_1,\dots,2\beta_n$, respectively. Suppose that

$$\mathbb{S}^{d-1}\setminus\left(\bigcup_{i=1}^n\mathbf{Z}_i\right)$$

consists of $2m$ spherically convex open connected components with inradii $F4D0\gamma_1,\dots,\gamma_{2m}$, respectively. Didid's conjecture. Then

$$2\beta_1+\dots+2\beta_n+\gamma_1+\dots+\gamma_{2m}\geq\pi.$$

This is a generalization of the zone-covering theorem: it relates the total widths of the zones and the inradii of the complementary components. The supplied text attributes the problem to Didid through Polyanskii's account, but gives no resolution.