  Measuring the magnetic field in cosmic filaments reveals how the Universe is
magnetised and the process that magnetised it. Using the Rotation Measures (RM)
at 144-MHz from the LoTSS DR2 data, we analyse the rms of the RM extragalactic
component as a function of redshift to investigate the evolution with redshift
of the magnetic field in filaments. From previous results, we find that the
extragalactic term of the RM rms at 144-MHz is dominated by the contribution
from filaments (more than 90 percent). Including an error term to account for
the minor contribution local to the sources, we fit the data with a model of
the physical filament magnetic field, evolving as $B_f = B_{f,0}\,(1+z)^\alpha$
and with a density drawn from cosmological simulations of five magnetogenesis
scenarios. We find that the best-fit slope is in the range $\alpha = [-0.2,
0.1]$ with uncertainty of $\sigma_\alpha = 0.4$--0.5, which is consistent with
no evolution. The comoving field decreases with redshift with a slope of
$\gamma = \alpha - 2 = [-2.2, -1.9]$. The mean field strength at $z=0$ is in
the range $B_{f,0}=39$--84~nG. For a typical filament gas overdensity of
$\delta_g=10$ the filament field strength at $z=0$ is in the range
$B_{f,0}^{10}=8$--26~nG. A primordial stochastic magnetic field model with
initial comoving field of $B_{\rm Mpc} = 0.04$--0.11~nG is favoured. The
primordial uniform field model is rejected.
