Wasserstein-$p$ local-dependence bound for normal approximation

Let $p$ be a positive integer and let $W=\sum_{i\in I}X_i$, where ${\mathbbm{E}}X_i=0$, ${\mathbbm{E}}W^2=1$, and the variables satisfy (LD1)--(LD$(p+1)$): for each $i_1\in I$, $i_2\in A_{i_1},\ldots,i_m\in A_{i_1\ldots i_{m-1}}$, there is a set $A_{i_1\ldots i_m}\supset A_{i_1\ldots i_{m-1}}$ such that $\\{X_{i_1},\ldots,X_{i_m}\\}$ is independent of $\\{X_j:j\notin A_{i_1\ldots i_m}}$. Define

$$R_m=\sum_{i_1\in I}\sum_{i_2\in A_{i_1}}\dots\sum_{i_{m+2}\in A_{i_1\dots i_{m+1}}}\sum_{({\mathbbm{E}})}{\mathbbm{E}}|X_{i_1}X_{i_2}|({\mathbbm{E}})|X_{i_3}|\cdots({\mathbbm{E}})|X_{i_{m+2}}|,$$

where $\sum_{({\mathbbm{E}})}$ is the sum over the possible placements of ${\mathbbm{E}}$ before each $X_i$, with any two ${\mathbbm{E}}$'s separated by at least two $X_i$'s. Wasserstein-$p$ conjecture. Under these assumptions,

$$\mathcal{W}_p(\mathcal{L}(W),n(0,1))\leqslant C_p\sum_{m=1}^p(R_m)^{1/m},$$

where $C_p$ depends only on $p$. This conjectured extension of the Wasserstein-1 normal-approximation bound concerns sums of locally dependent random variables. The case $p=1$ was proved by Barbour, Karoński and Ruciński in 1989; the general assertion is therefore resolved at least in the stated base case, while the supplied evidence does not establish the status for every positive integer $p$.