Let $k$ be a perfect field of characteristic $p>0$ and consider an ordinary liftable Calabi-Yau threefold $X_{0}/k$. By this I mean that $H^{i}(X_{0},B_{X_{0}/k}^{j})=0$ for all $i\geq 0$ and $j\geq 1$, where $B_{X_{0}/k}^{j}:=\text{im}(d:\Omega_{X_{0}/k}^{j-1}\rightarrow\Omega_{X_{0}/k}^{j})$, and $\mathfrak{X}/W(k)$ is a lifting of $X_{0}$ to characteristic zero.

Now, let $X/R$ be a proper smooth lifting of $X_{0}$ over an artinian local algebra $R$ and let $A$ be an artinian local $R$-algebra with residue field $k$. I would like to show that \begin{equation*} H_{\text{fl}}^{2}(X\otimes_{R}A,\mu_{p^{\infty}})=0 \end{equation*} I have two questions...

a) Is this true?

b) If so, how do I show it?

Remarks:-

1. This is true if instead $X_{0}$ was a K3 surface and I was looking at $H_{\text{fl}}^{1}(X\otimes_{R}A,\mu_{p^{\infty}})$.

2. Under our hypotheses on $X_{0}$, $Br(X_{0})$ has no $p$-torsion, so I can show that \begin{equation*} H_{\text{fl}}^{2}(X\otimes_{R}A,\mu_{p^{r}})\cong\frac{Pic(X\otimes_{R}A)}{Pic(X\otimes_{R}A)^{p^{r}}} \end{equation*} but I don't know how to attack the right hand side.

Thank you very much for your help, and please don't hesitate to point out mistakes or ask for clarifications.