  Context. Radio halos are megaparsec-scale diffuse radio sources{ mostly}
located at the centres of merging galaxy clusters. The common mechanism invoked
to explain their origin is the re-acceleration of relativistic particles caused
by large-scale turbulence. Aims. Current re-acceleration models predict that a
significant number of halos at high redshift should be characterised by very
steep spectra ($\alpha<-1.5$) because of increasing inverse Compton energy
losses. In this paper, we investigate the spectral index properties of a sample
of nine clusters selected from the second Planck Sunyaev-Zel'dovich catalogue
showing diffuse radio emission with the Low Frequency Array (LOFAR) in the
120-168 MHz band. This is the first time that radio halos discovered at low
frequencies are followed up at higher frequencies. Methods. We analysed
upgraded Giant Metrewave Radio Telescope (uGMRT) observations in Bands 3 and 4,
that is, 250-500 and 550-900 MHz respectively. These observations were combined
with existing LOFAR data to obtain information on the spectral properties of
the diffuse radio emission. Results. We find diffuse radio emission in the
uGMRT observations for five of the nine high-$z$ radio halos previously
discovered with LOFAR. For those, we measure spectral indices in the range of
$-1$ to $-1.4$. For the uGMRT non-detections, we estimated that the halos
should have a spectral index steeper than $-1.5$. We also confirm the presence
of one candidate relic. Conclusions. Despite the small number of clusters, we
find evidence that about half of the massive and merging clusters at high
redshift host radio halos with a very steep spectrum. This is in line with
theoretical predictions, although larger statistical samples are necessary to
test models.
