Tensor products of H-unital rings

Determine whether the tensor product of two H-unital nonunital E₁-rings is necessarily H-unital.

Background

An H-unital nonunital E₁-ring is one whose unitalization maps to the sphere spectrum by a homological epimorphism. The paper proves that tensoring an H-unital ring with the fiber of a map between idempotent sphere-algebras preserves H-unitality, and also proves the result for connective H-unital rings. However, the general closure of H-unital rings under tensor products is posed explicitly as an unresolved question.