Deterministic sub-distributive plasma conjecture

Let $R$ be a plasma, with multiplication map

$$\bar\wedge\colon R\wedge R\to R.$$

The monoidal structure is sub-distributive when

$$a\bar\wedge(b\boxplus c)\subseteq(a\bar\wedge b)\boxplus(a\bar\wedge c).$$

Let $\hat{H}R$ denote the associated $\mathbb{F}_1$-module.

Deterministic sub-distributive plasma conjecture. If $R$ is a deterministic plasma with a sub-distributive monoidal structure, then $\hat{H}R$ is canonically an $\mathbb{F}_1$-algebra.

This conjecture proposes a sufficient condition under which the plasmic nerve preserves enough monoidal structure to produce an $\mathbb{F}_1$-algebra. The source gives no resolution.