  The intergalactic medium (IGM) acts like a calorimeter recording energy
injection by cosmic structure formation, shocks and photoheating from stars and
active galactic nuclei. It was recently proposed that spatially inhomogeneous
TeV-blazars could significantly heat up the underdense IGM, resulting in
patches of both cold and warm IGM around $z\simeq2-3$. The goal of this study
is to compare predictions of different blazar heating models with recent
observations of the IGM. We perform a set of cosmological simulations and
carefully compute mock observables of the Lyman-$\alpha\ ($Ly$\alpha$) forest.
We perform a detailed assessment of different systematic uncertainties which
typically impact this type of observables and find that they are smaller than
the differences between our models. We find that our inhomogeneous blazar
heating model is in good agreement with the Ly$\alpha$ line properties and the
rescaled flux probability distribution function at high redshift ($2.5<z<3$)
but that our blazar heating models are challenged by lower redshift data
($2<z<2.5$). Our results could be explained by HeII reionisation although
state-of-the-art models fall short on providing enough heating to the
low-density IGM, thus motivating further radiative transfer studies of
inhomogeneous HeII reionisation. If blazars are indeed hosted by group-mass
halos of $2\times10^{13}{M}_\odot$, a later onset of blazar heating in
comparison to previous models would be favoured, which could bring our findings
here in agreement with the evidence of blazar heating from local gamma-ray
observations.
