So in the thought experiment Wigners friend the paradox is ultimately due to a difference of descriptions of density matrices.
If the physical variable that is measured of the spin system is denoted by $z$, where $z$ takes the possible outcome values $0$ or 1, the above Wigner's friend situation is modelled in the RQM context as follows: $F$ models the situation as the before-after-transition:
$$ \alpha | 0 \rangle_S + \beta | 1 \rangle_S \to |1 \rangle _S $$
of the state of $S$ relative to him (here it was assumed that $F$ received the outcome $z = 1$ in his measurement of $S$). In RQM language, the fact $z = 1$ for the spin of $S$ actualized itself relative to $EF$ during the interaction of the two systems.
A different way to model the same situation is again an outside (Wigner's) perspective. From that viewpoint, a measurement by one system ($F$) of another ($S$) results in a correlation of the two systems. The state displaying such a correlation is equally valid for modelling the measurement process. However, the system with respect to which this correlated state is valid changes. Assuming that Wigner ($W$) has the information that the physical variable $z$ of $S$ is being measured by $F$, but not knowing what $F$ received as result, $W$ must model the situation as:
$$ (\alpha |0 \rangle_S + \beta |1 \rangle_S) | \perp \rangle_F \to \alpha ( | 0 \rangle_S \otimes | 0 \rangle_F ) + \beta (|1\rangle_S \otimes |1 \rangle_F) $$
where ${F}$ is considered the state of $F$ before the measurement, and ${\displaystyle |1\rangle _{F}}{\displaystyle |1\rangle _{F}}$ and ${\displaystyle |0\rangle _{F}}{\displaystyle |0\rangle _{F}}$ are the states corresponding to $F$'s state when he has measured $1$ or $0$ respectively. This model is depicting the situation as relative to $W$, so the assigned states are relative states with respect to the Wigner system. In contrast, there is no value for the $z$ outcome that actualizes with respect to $W$, as he is not involved in the measurement.
The assumptions boil down to those assumptions are
(Q): Quantum theory is correct. (C): Agent's predictions are information-theoretically consistent. (S): A measurement yields only one single outcome.
Now, while decoherence does not solve the measurement problem it everyone seems to agree that it is a step in the right direction. Does de-coherence add value to the discussion when people question axiom $(S)$?
To make more explicit the time evolution can be reformulated in terms of density matrices to remove axiom C. Let's say my wavefunction is in superposition of
$$|\psi \rangle_S = \frac{1}{\sqrt{2}}(\alpha | 0 \rangle_S + \beta | 1 \rangle_S)$$
The corresponding density matrix is:
$$ \rho_S = \begin{bmatrix} |\alpha|^2 & \alpha^* \beta \\ \beta^* \alpha & |\beta|^2 \end{bmatrix} $$
Now, the time average of the density matrix is decoherence of the density matrix which happens due to decoherence is given by:
$$ \langle \rho_S \rangle_t = \begin{bmatrix} |\alpha|^2 & 0 \\ 0 & |\beta|^2 \end{bmatrix} $$
And proceeds to use the density matrix $\rho'$
$$\rho' = \langle \rho_S \rangle_t \tag{D}$$
Via this maneuver I reformulate the problem and still get the contradiction $\rho' \neq \rho$ (replacing axiom (S) with (D)). To me its very obvious one of them is using approximations statement $(D)$. More explicitly:
One of them is using:
$$i \hbar \frac{\partial \rho }{\partial t} = [H,\rho]$$
and the other:
$$i \hbar \frac{\partial \langle \rho \rangle_t}{\partial t} = [H,\langle \rho \rangle_t]$$
The real problem I encounter when I try to explain myself is $(D)$ is a valid replacement of $(S)$. What happens if I put both Wigner and his friend in box (the entire setup).
Edit: In response to Mateus Araújo's answer:
"if the description of the measurement becomes the same if the Friend's decoherence reaches Wigner, i.e., if the Friend is not inside a decoherence-proof box."
Yes, this does happen. Once, decoherence reaches the Wigner. Then Wigner is also using the equation:
$$i \hbar \frac{\partial \langle \rho \rangle_t}{\partial t} = [H,\langle \rho \rangle_t]$$
As a result neither of them see any discrepancy now for both of them their descriptions of the density matrices match.