The prequiver form conjecture for knots-quivers correspondence

Let $K$ be a knot admitting the standard knots-quivers correspondence, so that the node variables satisfy $x_i\sim x$. Let $Q$ be a quiver corresponding to $K$, with one spectator node and $k$ paired blocks. Prequiver form conjecture. There exists a unique such quiver $Q$ whose adjacency matrix has the block form

$$C=\begin{pmatrix}0&F_1&F_2&\cdots&F_k\F_1^T&D_1&U_{12}&\cdots&U_{1k}\F_2^T&U_{12}^T&D_2&\cdots&U_{2k}\vdots&\vdots&\vdots&\ddots&\vdots\F_k^T&U_{1k}^T&U_{2k}^T&\cdots&D_k\end{pmatrix},$$

where

$$F_i=\begin{bmatrix}\check C_{0i}&\check C_{0i}\end{bmatrix},\qquad D_i=\begin{bmatrix}\check C_{ii}&\check C_{ii}\check C_{ii}&\check C_{ii}+1\end{bmatrix},$$ $$U_{ij}=\begin{bmatrix}\check C_{ij}&\check C_{ij}\check C_{ij}+1&\check C_{ij}+1\end{bmatrix},\qquad \check C_{mn}\in\mathbb{Z}.$$

The block structure encodes the spectator node and paired nodes associated with $\mathfrak{sl}_1$ cancellation; the source gives no resolution evidence.