open quantum systems - Is hermicity of the reduced density matrix preserved here?

I am following along Breuer and Petruccione's book . I would like to know if the property $\rho^{\dagger} = \rho$ is preserved for evolution that is described by the Born Approximation.

For a Hilbert space $\mathscr{H}_s \otimes \mathscr{H}_{b}$ describing a system of interest and some reservoir/bath, we consider a time-independent Hamiltonian of the form

$$H = H_s \otimes \mathbb{I}_b + \mathbb{I}_s \otimes H_b + g H_{int}$$ where $g$ is some small coupling and $H_{int}$ describes some interaction between the system and the bath.

The full density matrix of the combined system and bath $\sigma(t)$ evolves via the von Neumann equation

$$\frac{d\sigma_I(t)}{dt} = - i [ V(t) , \sigma_{I}(t) ]$$ which is written in the interaction-picture, where $$\sigma_{I}(t) := e^{+ i H_s t} \otimes e^{+ i H_b t} \sigma(t) e^{- i H_s t} \otimes e^{- i H_b t} \ \ \ \ \text{and} \ \ \ \ V(t) := e^{+ i H_s t} \otimes e^{+ i H_b t} H_{int} e^{- i H_s t} \otimes e^{- i H_b t}$$

If we define the reduced density matrix describing the system as the following partial trace

$$\rho(t) := \mathrm{Tr}_{b}[ \sigma(t) ]$$ and then $\rho_{I}(t) := e^{+ i H_s t} \rho(t) e^{- i H_s t}$, the Born Approximation says that $$\frac{d\rho_I(t)}{dt} \simeq - g^2 \int_0^t ds\ \mathrm{Tr}_b\bigg( \big[ V(t), [V(s), \rho_I(s) \otimes \varrho_{B}] \big] \bigg)$$ where $\varrho_b$ is the initial state of the bath at $t=0$, where $\sigma(0) = \rho(0) \otimes \varrho_b$.

My Question: Suppose that $\rho(0)^{\dagger}= \rho(0)$ so that the initial density matrix is Hermitian. How can you use the above equation to show that $\rho(t)^{\dagger}= \rho(t)$ for $t>0$?

Is this possible? In the literature, I sometimes come across some statements that Lindblad equations preserve Hermicity (Lindblad equations being the above equation, after taking the Markov Approximation and then the secular/rotating-wave approximation).

Is it possible to prove this from this equation of motion? Or do we need additional assumptions? Or can this be more generally proven for $\rho$ without specifying to a particular evolution equation?