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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-$α ($Ly$α$) 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$α$ 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.