  Core-collapse supernovae (SNe) are believed to be the first significant
source of dust in the Universe. Such SNe are expected to be the main dust
producers in young high-redshift Lyman $\alpha$ emitters (LAEs) given their
young ages, providing an excellent testbed of SN dust formation models during
the early stages of galaxy evolution. We focus on the dust enrichment of a
specific, luminous LAE (Himiko, $z\simeq 6.6$) for which a stringent upper
limit of $52.1~\mu$Jy ($3\sigma$) has recently been obtained from ALMA
continuum observations at 1.2 mm. We predict its submillimetre dust emission
using detailed models that follow SN dust enrichment and destruction and the
equilibrium dust temperature, and obtain a plausible upper limit to the dust
mass produced by a single SN: $m_\mathrm{d,SN} < 0.15$--0.45 M$_\odot$,
depending on the adopted dust optical properties. These upper limits are
smaller than the dust mass deduced for SN 1987A and that predicted by dust
condensation theories, implying that dust produced in SNe are likely to be
subject to reverse shock destruction before being injected into the
interstellar medium. Finally, we provide a recipe for deriving
$m_\mathrm{d,SN}$ from submillimetre observations of young, metal poor objects
wherein condensation in SN ejecta is the dominant dust formation channel.
