Doubly distributive hyperfields satisfy the inheritance property

Let $p\in\mathbb{H}[X]$ be a polynomial, and let $A_p$ be its list of roots, including repetitions corresponding to multiplicities. A hyperfield has the inheritance property at $p$ if, for every subset $\{a_{j_1},\dots,a_{j_m}\}\subseteq A_p$ with $m\leq\operatorname{deg}(p)$, there exists $q\in\mathbb{H}[X]$ such that

$$p\in (X-a_{j_1})\odot(X-a_{j_2})\odot\dots\odot(X-a_{j_m})\odot q.$$

It has the inheritance property in general if this holds for every polynomial $p\in\mathbb{H}[X]$.

Inheritance-property conjecture. All hyperfields with the doubly distributive property satisfy the inheritance property.

The inheritance property is introduced to address the non-uniqueness of polynomial factorisations over hyperfields, even for doubly distributive hyperfields. The paper presents this as an open direction needed to extend the field-theoretic factorisation arguments underlying its version of Kapranov's theorem.