Surgery-limit invariants for $\mathit{HP}$ and $\mathit{HP}_{\#}$

Let $K\subset S^3$ be a knot, let $p$ be fixed, and let $S^3_{p/q}(K)$ denote $p/q$ surgery, with $q$ tending to infinity through integers relatively prime to $p$. For each degree $n$, consider the normalized ranks of the two Floer theories $\mathit{HP}$ and $\mathit{HP}_{\#}$. Surgery-limit invariant conjecture. The quantities

$$\lim\limits_{q\to\infty} \frac{1}{q}\operatorname{rk}(\mathit{HP}^n(S^3_{p/q}(K)))$$

and

$$\lim\limits_{q\to\infty} \frac{1}{q}\operatorname{rk}(\mathit{HP}_{\#}^n(S^3_{p/q}(K)))$$

are well-defined invariants of the knot $K$. This is proposed as the analogue for $\mathit{HP}$ and $\mathit{HP}_{\#}$ of the $\operatorname{SL}(2,\mathbb{C})$ Casson knot invariant; the source does not provide a resolution.