Doubly distributive hyperfields satisfy the multiplicity bound

A hyperfield is an algebraic structure with multivalued addition and multiplication satisfying the hyperfield axioms. A hyperfield has the doubly distributive property when its operations satisfy the corresponding strengthened distributive law. The multiplicity bound is the property that the multiplicity of a root of a polynomial is bounded by the degree of that polynomial.

Multiplicity-bound conjecture. All hyperfields with the doubly distributive property satisfy the multiplicity bound.

The paper identifies the multiplicity bound as one of the properties of fields needed for its generalisation of Kapranov's theorem to hyperfields. It states this as a conjectural property of doubly distributive hyperfields, including the tropical, sign, and Krasner hyperfields, but does not establish it in general.