Numerical exceptional collection conjecture for fake projective spaces

Let $M$ be an $n$-dimensional fake projective space whose canonical class is numerically divisible by $n+1$. Choose a line bundle $L$ such that

$$K_M=(n+1)L,$$

and let $E_i$ be line bundles satisfying

$$E_i\equiv-iL, 1\leq i\leq n.$$

Numerical exceptional collection conjecture. For a suitable choice of the line bundles $E_i$, the sequence

$$\mathcal{O}_M,E_1,E_2,\dots,E_n$$

is an exceptional collection on $M$.

This is proposed as a more general problem motivated by the search for exceptional objects on fake projective spaces. The supplied text gives no resolution status; the problem is presented as a conjectural extension of the ordinary divisibility statement.