Conjecture on improving the Poisson approximation bound in Wasserstein distance

Let $X_i$, $1\leqslant i\leqslant n$, be independent Bernoulli random variables with expectations $\mathbb{E}X_i=p_i$. Define $W=\sum_{i=1}^n X_i$, $\mu=\sum_{i=1}^n p_i$, $\mu_l=\sum_{i=1}^n p_i^l$, and $\lambda=\mu-\mu_2$, and suppose that $\mu_2$ is an integer. Let $\pi$ denote the Poisson distribution with mean $\lambda$, and let $\mathbb{W}_2$ denote the Wasserstein distance of order $2$. The preceding bound is

$$\mathbb{W}_2(\mathcal{L}(W),\pi*\delta_{\mu_2})=\mathbb{W}_2(\mathcal{L}(W-\mu_2),\pi)\leqslant \mu_2e^{-\lambda^2/(4\mu)}+\left\{6(\mu_2-\mu_3)+\mu_2(7+\lambda)e^{-\lambda^2/(2\mu)}\right\}^{1/2}.$$

Conjecture on improving the Poisson approximation bound. The order of the upper bound in the displayed inequality can be significantly improved.

The conjecture concerns the accuracy of Poisson approximation for Poisson binomial distributions under the quadratic Wasserstein distance. The source gives no precise proposed order or evidence resolving the question, so its status remains open.