Composite quantum Chernoff conjecture for finitely many hypotheses

Let $\varrho$ and $\sigma_1,\ldots,\sigma_r$ be quantum states, and define the sequences

$$\vec \varrho:=\{\varrho^{\otimes n}\}_{n\in\mathbb{N}},\qquad \vec \sigma_i:=\{\sigma_i^{\otimes n}\}_{n\in\mathbb{N}}\quad (i\in[r]).$$

For the optimal worst-case composite error probability, write $p_e(\vec \varrho,\{\vec\sigma_1,\ldots,\vec\sigma_r\})$ for the asymptotic error exponent when its limit exists.

Composite quantum Chernoff conjecture. The limit $p_e(\vec \varrho,\{\vec\sigma_1,\ldots,\vec\sigma_r\})$ exists, and

$$p_e(\vec \varrho,\{\vec\sigma_1,\ldots,\vec\sigma_r\})=\max_i p_e(\vec \varrho,\vec\sigma_i).$$

Equivalently, the composite exponent is determined by the least favorable individual hypothesis, or by the minimum of the corresponding Chernoff divergences. The conjecture is open even when $r=2$, with one state in the first hypothesis and two states in the second.