Non-stochastic matrix verifying detailed balance equations. Is it diagonalizable?

Question

1. Let $P$ be a real square matrix. $P=[P_{ij}]\in\mathbb{R}^{n\times n}$.
2. Let $\pi$ be a positive real vector. $\pi=[\pi_j]\in\mathbb{R}^n: \pi_j>0$.
3. Let $P$ satisfy the detailed balance equations with $\pi$. $\pi_i P_{ij}=\pi_j P_{ji}$.

Q: Is $P$ diagonalizable?

If $P$ is a stochastic matrix, than the answer is yes.

However, to my understanding, the linked proof never uses the fact that $P$ is stochastic, or that $\pi$ is the only eigenvector of $P$ with unit eigenvalue.

They show that the detailed balance equations (3.) imply that $P$ is similar to the symmetric $\big(D^\frac{1}{2} PD^\frac{-1}{2}\big)$, with $D := diag(\mathbf \pi)$. So, as long as $\pi_j^{-\frac12}$ is well defined, the proof should hold. Did I miss something?

Question: Do conditions 1., 2., and 3. imply that $P$ is diagonalizable?