Quadratic-time algorithm for overlapping Hankel block low-rank completion

Consider the overlapping Hankel block low-rank completion problem, with matrix-size parameter $N$, and let $\hat{X}$ denote a candidate completed matrix. Quadratic-time completion conjecture. There exists an algorithm that computes a candidate solution $\hat{X}$ to the overlapping Hankel block low-rank completion problem in

$$\mathcal{O}(N^2)$$

time. The current approach has worst-case complexity $\mathcal{O}(N^4)$, while the authors explain that intermingling its two stages may reduce the complexity to $\mathcal{O}(N^2)$; the conjecture remains unresolved in the source.