Ternary Egyptian fractions with prime denominator

For a prime number $p$, let $A_3(p)= | { m \in $\mathbb{N}$: \exists m_1,m_2,m_3 \in $\mathbb{N}$, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} } |$. In 2019 Luca and Pappalardi proved that $x (\log x)^3 \ll \sum_{p \le x} A_{3}(p) \ll x (\log x)^5$\mathrm{.}Weimprovetheupperbound,showing$\sum_{p \le x} A_{3}(p) \ll x (\log x)^3 (\log \log x)^2$.