Path-category realization conjecture for spectroids

Let $\Bbbk$ be a field, and let $\mathcal{C}$ be a spectroid, meaning a small $\Bbbk$-linear category whose Hom-spaces are finite-dimensional and whose nonzero endomorphism algebras are division algebras. A path category is constructed from a pair $(\mathbb{X},\Gamma{/}{\sim})$ by taking objects $\mathbb{X}$ and morphisms represented by equivalence classes of paths; its $\Bbbk$-linear path category is the $\Bbbk$-linearization of this category, with a zero object.

Path-category realization conjecture. For every spectroid $\mathcal{C}$, there exists a choice of $(\mathbb{X},\Gamma{/}{\sim})$ such that $\mathcal{C}$ is the $\Bbbk$-linear path category of $(\mathbb{X},\Gamma{/}{\sim})$.

The conjecture asks whether every spectroid admits a realization by a path category and its $\Bbbk$-linearization. The paper states that this question falls outside its scope, and no resolution is supplied here.