Obtain Mean Field Equations for Spin Models using a uniform Ansatz

I would like to see how my model I am working on behaves in the limit of infinite dimensions so I get a little bit of intuition for the low dimensional case. In the paper I am reading they have a transverse Ising model:

$
H = -J sum_{<i,j>}sigma_i^x sigma_j^x+Delta sum_i sigma_i^z
$

with an additional way of spontaneous emission with Lindblad operators $L_i = sqrt{Gamma} sigma_i^-$ so the dynamics are governed by the master equation.
They just take it for granted that using a "uniform ansatz" one gets the MF equation:

$
frac{d}{dt} X = – Delta Y – frac{Gamma}{2} X
$

where $X=langle sigma_x rangle$.

What does this uniform ansatz mean in this context? I only know that MF usually means that one replaces the spins with the average magnetization and as such can reduce the problem to an easier one where the spins only feel an effective magnetic field.

Or is it really easy to obtain this equation?