Bounded complement approximation conjecture for delta invariants

Let $(X,\Delta)$ be a log Fano pair, meaning that $(X,\Delta)$ is log Fano, and suppose that $\delta(X,\Delta)\leq 1$. Let $A_{X,\Delta}(E)$ denote the log discrepancy of a prime divisor $E$ over $X$, $S_{X,\Delta}(E)$ its expected vanishing order, and let an $N$-complement of plt type $(X,\Delta_i^+)$ be a complement of plt type with complement index $N$. An lc place of $(X,\Delta_i^+)$ is a divisor defining a log canonical place of that pair.

Bounded complement approximation conjecture. There is a natural number $N$ depending only on $(X,\Delta)$ and a sequence of prime divisors $\{E_i\}$ over $X$ such that

$$\delta(X,\Delta)=\lim_i\frac{A_{X,\Delta}(E_i)}{S_{X,\Delta}(E_i)},$$

and, for each $E_i$, one can find an $N$-complement of plt type $(X,\Delta_i^+)$ such that $E_i$ is an lc place of $(X,\Delta_i^+)$.

The preceding theorem establishes the analogous statement without requiring a bounded complement index. This conjecture asks for a single index $N$, depending only on the fixed pair, that works for every divisor in the approximating sequence.