Let $C(e^{i\theta})$ be an $m\times n$ ($m\ge n$) matrix-valued continuous function of $\theta\in[-\pi,\pi]$. Let $A_1(e^{i\theta})$ and $A_2(e^{i\theta})$ be two $n\times n$ positive definite matrix-valued continuous function of $\theta\in[-\pi,\pi]$ (i.e. $A_1(e^{i\theta}),\,A_2(e^{i\theta})>0$, $\forall\,\theta\in[-\pi,\pi]$) satisfying $A_1(e^{i\theta})\neq A_2(e^{i\theta})$. Suppose that it holds $$\tag{1}\label{1} \int_{-\pi}^{\pi} C(e^{i\theta})A_1(e^{i\theta})C^*(e^{i\theta})\,\mathrm{d}\theta\neq \int_{-\pi}^{\pi} C(e^{i\theta})A_2(e^{i\theta})C^*(e^{i\theta})\,\mathrm{d}\theta, $$ where the superscript $*$ denotes the Hermitian transpose.
My Question: Let $P(e^{i\theta})$ be any positive definite $n\times n$ continuous function of $\theta\in[-\pi,\pi]$, does \eqref{1} imply that $$\tag{2}\label{2} \int_{-\pi}^{\pi} CA_1^{1/2}P A_1^{1/2}C^*\,\mathrm{d}\theta\neq \int_{-\pi}^{\pi} CA_2^{1/2}P A_2^{1/2}C^*\,\mathrm{d}\theta\ \ \text{?} $$
N.B. In Eq. \eqref{2} I omitted the dependence on $\theta$ for readability. Moreover, the superscript ${1/2}$ denotes the principal matrix square root.
Note. The answer is negative, as a counterexample posted in a comment by Pietro Majer shows. However, it is not still clear if the answer is negative if $A_1$ and $A_2$ are taken of the form $$ A_1(e^{i\theta})=(C^*(e^{i\theta}) X_1 C(e^{i\theta}))^{-1},\quad A_2(e^{i\theta})=(C^*(e^{i\theta}) X_2 C(e^{i\theta}))^{-1} $$ where $X_1$ and $X_2$ are constant $n\times n$ positive matrices and $C$ is of full column rank a.e..
