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\Bigl\lvert\sum_{j=1}^n X_j\Bigr\rvert^p\le C\sum_{j=1}^n E\lvert X_j\rvert^p \end{equation} for all natural $n$ and all pairwise independent zero-mean real-valued random variables $X_1,\dotsc,X_n$.

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

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 $X_j$ 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, and a positive answer to Q2 would imply a negative answer to Q1.

By Proposition 1.8 (iii), letting $D_p$ denote the smallest constant factor $C$ in \eqref{eq:pair} for all natural $n$ and all completely independent zero-mean real-valued random variables $X_1,\dotsc,X_n$, we have $D_p\to1$ as $p\uparrow2$.