The extremal orbifold Betti-number and Euler-characteristic conjecture

Let $n$ be a positive integer, and consider projective $n$-dimensional varieties with quotient singularities and trivial canonical class. Write $H$ for the sum of their orbifold Betti numbers. For odd $n$, write $\chi_{\mathrm{orb}}$ for the orbifold Euler characteristic. Extremal orbifold invariants conjecture. In dimension $n$, the largest possible value of $H$ is

$$H=2(s_0-1)\cdots(s_n-1).$$

For odd $n$, the smallest possible orbifold Euler characteristic is

$$-(s_0-1)\cdots(s_{n-1}-1)(2s_n-6),$$

and the largest is

$$(s_0-1)\cdots(s_{n-1}-1)(2s_n-6).$$

The conjecture is motivated by three explicit examples whose common Betti-number sum is the largest known for Calabi-Yau threefolds. The source gives no resolution status.