Palindromic factorization of rich words

A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. For every finite rich word $w$ there are distinct nonempty palindromes $w1, w2,\dots,wp$ such that $w=wpw{p-1}\cdots w1$ and $wi$ is the longest palindromic suffix of $wpw{p-1}\cdots wi$, where $1\leq i\leq p$. This palindromic factorization is called \emph{UPS-factorization}. Let $luf(w)=p$ be \emph{the length of UPS-factorization} of $w$. In 2017, it was proved that there is a constant $c$ such that if $w$ is a finite rich word and $n=\vert w\vert$ then $luf(w)\leq c\frac{n}{\ln{n}}$. We improve this result as follows: There are constants $\mu, \pi$ such that if $w$ is a finite rich word and $n=\vert w\vert$ then [luf(w)\leq \mu\frac{n}{e^{{\pi\sqrt{\ln{n}}}}\mbox{.}]} The constants $c,\mu,\pi$ depend on the size of the alphabet.