Threshold-policy conjecture for the greedy observation policy

For an information state $z\in[0,1]$, let $\alpha_i(z)$ be the probability of selecting observation process $i$ and let $r_i(z)$ be the resulting information state, with entropy $H$. The greedy policy minimises the expected entropy after one observation by choosing process $0$ when

$$\alpha_0(z)H(r_0(z))<\alpha_1(z)H(r_1(z)),$$

and process $1$ otherwise, up to exchanging strict and non-strict inequalities.

Greedy threshold-policy conjecture. The greedy policy is always a threshold policy.

The claim is supported in the excerpt only by computational results: across 152000 sampled parameter values, the defining functions crossed at most once. A proof is not given.