What exactly is meant by a "time-reversed Hamiltonian"

For context, I am reading this paper.

Basically, the paper makes reference to “evolving with respect to the time-reversed Hamiltonian”. I’m slightly unclear as to what this actually means. Here is my logic:

Let $H$ be some Hamiltonian with eigenstates $|E_nrangle$. Let $H’$ be the time-reversed Hamiltonian, with eigenstates $|E_n’rangle$, where $|E_n’rangle = Theta |E_nrangle$ with $Theta$ being the time-reversal operator:

$$Theta = i Y K$$

where $K$ is the complex conjugation operator, and $Y$ is the Pauli-Y. It follows that $H’$ can be written as:

$$H’ = -Theta H’ Theta$$

Since, given some $|E_n’rangle$, we will have:

$$H’ |E_n’rangle = -Theta H Theta |E_n’rangle = Theta H |E_nrangle = Theta E_n |E_nrangle = E_n |E_n’rangle$$

So the time-reversed states are eigenstates of this Hamiltonian. In addition, energy doesn’t change under time-reversal, and as you can see, we retain the same energy $E_n$ for a given eigenstate under time-reversal, thus we have uniquely defined the time-reversed Hamiltonian.

I then went on to define time-evolution under this Hamiltonian:

$$e^{-i gamma H’} = e^{i gamma Theta H Theta} = displaystylesum_{n = 0}^{infty} frac{(i gamma Theta H Theta)^n}{n!}$$

We know that $-Theta^2 = mathbb{I}$, as time-reversing a quantum state twice is equivalent to doing nothing. Therefore, all of the middle terms in the product expansion of the numerator of each term will cancel, and we will have:

$$displaystylesum_{n = 0}^{infty} frac{(i gamma Theta H Theta)^n}{n!} = displaystylesum_{n = 0}^{infty} (-i Y K) frac{(-i gamma H )^n}{n!} (i Y K) = Y K Big( displaystylesum_{n = 0}^{infty} frac{(-i gamma H )^n}{n!} Big) Y K = Y K e^{-i gamma H} Y K$$

This clearly makes no sense: time-evolution must be unitary and $K$ is anti-unitary.

I highly doubt that my thinking is correct, as I watched a talk given by one of the authors of the paper, and he said that the time-reversed Hamiltonian is “usually” the same as the original Hamiltonian. Where am I going wrong?