Residual representability of the Heegaard category

Let ${\cal H}$ be the Heegaard category, and let $T(V)$ denote the relevant target associated with a vector space $V$. Residual representability conjecture. If $a$ and $b$ are distinct elements of ${\cal H}$, there exists a vector space $V$ and a morphism

$$m:{\cal H}\to T(V)$$

such that $m(a)\ne m(b)$. This would provide a computationally effective separation of distinct elements of the Heegaard category, strengthening its role in distinguishing closed $3$-manifolds; the supplied text does not indicate whether the conjecture has been resolved.