Square-root upper bound conjecture for the length of UPS-factorizations

Let $\operatorname{A}$ be a finite alphabet. For a finite nonempty rich word $w$ over $\operatorname{A}$, write its UPS-factorization as $w=w_pw_{p-1}\cdots w_1$, where the factors are palindromes, and let $\operatorname{luf}(w)=p$ be its length. Set $n=|w|$. Square-root UPS-factorization conjecture. There is a positive real constant $\lambda$ such that, for every finite nonempty rich word $w$ over $\operatorname{A}$,

$$\operatorname{luf}(w)\leq \lambda\sqrt{n}.$$

This would improve the paper's proved upper bound $\operatorname{luf}(w)\leq \mu n/e^{\pi\sqrt{\ln n}}$ and remains presented as a conjecture in the source.