If a convex combination of conformal matrices is conformal, are they all proportional?

Question

$\newcommand{\CO}{\text{CO}}$ $\newcommand{\SO}{\text{SO}}$ $\newcommand{\dist}{\text{dist}}$

Let $\CO(2) =\{\lambda R : R \in \SO(2)\, | \, \lambda > 0\} $ be the set of $2 \times 2$ conformal matrices.

Let $A_i \in \CO(2)$ be a finite number of conformal matrices, and suppose that $\sum \lambda_i A_i \in \CO(2)$, where $\lambda_i \ge 0$ and $\sum \lambda_i=1$.

Is it true that all the $A_i$ are multiples of the same conformal matrix?

I don't even know the answer for the special case where we have only two matrices in the convex combination.

Answer 1

Counterexample:

$$\frac{1}{2}\begin{pmatrix}1&0\\0&1\end{pmatrix}+\frac{1}{2}\begin{pmatrix}0&-1\\1&0\end{pmatrix}=\frac{1}{\sqrt2}\begin{pmatrix}\frac{1}{\sqrt2}&-\frac{1}{\sqrt2}\\\frac{1}{\sqrt2}&\frac{1}{\sqrt2}\end{pmatrix}$$