For $p\in(1,2)$, let $C_p$ be the smallest constant factor $C$ in the von Bahr--Esseen-type inequality 
\begin{equation}\label{eq:pair}\tag{1}
	E\Big|\sum_{j=1}^n X_j\Big|^p\le C\sum_{j=1}^n E|X_j|^p  
\end{equation}
for all **pairwise** independent zero-mean real-valued random variables $X_1,\dots,X_n$. 

According to [Theorem 4.4][1], 
\begin{equation}\label{eq:C<}
	C_p\le\frac4{2-p}.   
\end{equation}

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Questions: 

>**Q1:** Is it true that $\sup_{p\in(1,2)}C_p<\infty$?

>**Q2:** Is it true that $\inf_{p\in(1,2)}\big((2-p)C_p\big)>0$? 

>**Q3:** Is there a simply described asymptotic behavior of $C_p$ as $p\uparrow2$? 

>**Q3a:** Is there a simply described asymptotic behavior of the smallest constant factor $C$ in \eqref{eq:pair} as $p\uparrow2$ assuming also that each $X_i$ is symmetrically distributed? 

>Any correct and complete answer to any one of these four questions will be considered a correct and complete answer to this entire post. 

Of course, an answer to Q3 or Q3a will also be an answer to Q1 and Q2. Also, a positive  answer to Q1 would imply a negative answer to Q2.  


  [1]: https://arxiv.org/abs/2210.04391