Polylogarithmic cluster-size extension of the hypergraph bandwidth theorem

Let $H$ be the blow-up host in the hypergraph bandwidth theorem, with clusters denoting the vertex classes used in the blow-up construction. Write $\operatorname{poly}(\log n)$ for a polynomial bound in $\log n$.

Cluster-size conjecture. Theorem~ remains valid for $H$ with clusters of size at most

$$\operatorname{poly}(\log n).$$

The source explains that the current bound is only $\operatorname{poly}(\log\log n)$ and expects this improvement from the hypergraph extension of the Erdős–Stone theorem. The conjecture remains open.