First I will give some background of Quantum Cramer-Rao bound. There is an amount called Fisher Information:$F(\lambda)=\sum_x{p\left( x|\lambda \right) \left( \partial _{\lambda}\ln p\left( x|\lambda \right) \right) ^2}$ where $p\left( x|\lambda \right)$ is probability distribution of result $x$ based on a to be estimated parameter $\lambda$. We can translate this into quantum formalism as follows.
In quantum mechanics, according to the Born rule we have $p(x \mid \lambda)=\operatorname{Tr}\left[\Pi_{x} \varrho_{\lambda}\right]$ where $\left\{\Pi_{x}\right\}, \int d x \Pi_{x}=\mathbb{I}$, are the elements of a positive operator-valued measure $(\mathrm{POVM})$ and $\varrho_{\lambda}$ is the density operator parametrized by the quantity we want to estimate. Introducing the Symmetric Logarithmic Derivative (SLD) $L_{\lambda}$ as the selfadjoint operator satistying the equation $$ \frac{L_{\lambda} \varrho_{\lambda}+\varrho_{\lambda} L_{\lambda}}{2}=\frac{\partial \varrho_{\lambda}}{\partial \lambda} $$ we have that $\partial_{\lambda} p(x \mid \lambda)=\operatorname{Tr}\left[\partial_{\lambda} \varrho_{\lambda} \Pi_{x}\right]=\operatorname{Re}\left(\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x} L_{\lambda}\right]\right) .$ The Fisher Information(similar to discrete case given at the beginning) is then rewritten as $$ F(\lambda)=\int d x \frac{\operatorname{Re}\left(\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x} L_{\lambda}\right]\right)^{2}}{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]} $$ For a given quantum measurement, i.e. a POVM $\left\{\Pi_{x}\right\}$, Fisher Information establish the classical bound on precision, which may be achieved by a proper data processing, e.g. by maximum likelihood, which is known to provide an asymptotically efficient estimator. On the other hand, in order to evaluate the ultimate bounds to precision we have now to maximize the Fisher information over the quantum measurements. We have $$ \begin{aligned} F(\lambda) & \leq \int d x\left|\frac{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x} L_{\lambda}\right]}{\sqrt{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]}}\right|^{2} \\ &=\int d x\left|\operatorname{Tr}\left[\frac{\sqrt{\varrho_{\lambda}} \sqrt{\Pi_{x}}}{\sqrt{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]}} \sqrt{\Pi_{x}} L_{\lambda} \sqrt{\varrho_{\lambda}}\right]\right|^{2} \\ & \leq \int d x \operatorname{Tr}\left[\Pi_{x} L_{\lambda} \varrho_{\lambda} L_{\lambda}\right] \\ &=\operatorname{Tr}\left[L_{\lambda} \varrho_{\lambda} L_{\lambda}\right]\\ &=\operatorname{Tr}\left[\varrho_{\lambda} L_{\lambda}^{2}\right] \end{aligned}\tag{1} $$ On the other hand, the second inequality above is based on the Schwartz inequality $\left|\operatorname{Tr}\left[A^{\dagger} B\right]\right|^{2} \leq \operatorname{Tr}\left[A^{\dagger} A\right] \operatorname{Tr}\left[B^{\dagger} B\right]$ applied to $A^{\dagger}=\sqrt{\varrho_{\lambda}} \sqrt{\Pi_{x}} / \sqrt{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]}$ and $B=\sqrt{\Pi_{x}} L_{\lambda} \sqrt{\varrho_{\lambda}}$ and it is saturated when $$ \frac{\sqrt{\Pi_{x}} \sqrt{\varrho_{\lambda}}}{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]}=\frac{\sqrt{\Pi_{x}} L_{\lambda} \sqrt{\varrho_{\lambda}}}{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x} L_{\lambda}\right]} \quad \forall \lambda, $$ This condition is satisfied iff $\left\{\Pi_{x}\right\}$ is made by the set of projectors over the eigenstates of $L_{\lambda}$, which, in turn, represents the optimal POVM to estimate the parameter $\lambda .$
Then I will give a detailed description of my specific question. In this literature, the author states that
For the single-parameter quantum estimations, the quantum Cramér–Rao bound can be attained with a theoretical optimal measurement.
But it seems that the optimal measurement stated in the background might depend on parameter $\lambda$ while in the background part we see that we use $\partial_{\lambda} p(x \mid \lambda)=\operatorname{Tr}\left[\partial_{\lambda} \varrho_{\lambda} \Pi_{x}\right]$ which imply that $\Pi_x$ is independent of $\lambda$. So based on the knowledge of the background, I can't see why we can always find a parameter independent POVM $\Pi_x$ to saturate the inequality(1)?