Let $W_n$ be a set of a log pair having the following property:

For any $(X, D) \in W_n$

(1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K_X = 0$). Moreover, $X$ is a $\mathbb{Q}$-factorial variety with canonical singularities. (one can also assume $(X, \frac{1}{2} D)$ is klt).

(2) $D$ is an **integral Weil divisor** which is **big**.

Is it true that there exists a universal $\delta > 0$ (i.e. only depends on $n$), such that we can decompose $D$ in $\mathbb{Q}$-linearly equivalent as follows:
$$D \sim_\mathbb{Q} A + E,$$ where $A$ is a nef and big divisor with **coefficients** bigger than $ \delta$, and $E$ is an effective divisor?

Any suggestions related to the topic are welcome!!