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: $m_\mathrmd,SN < 0.15$--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 $m_\mathrmd,SN$ from submillimetre observations of young, metal poor objects wherein condensation in SN ejecta is the dominant dust formation channel.
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