Linear-time positive definite Hankel factorization conjecture

Let $\mathbf{H} \in \mathbb{R}^{n\times n}$ be a positive definite Hankel matrix with bit complexity $\ell$. A symmetric positive factorization algorithm should find a representation of a matrix $\mathbf{B}$ with $n$ rows, $\tilde{O}(n)$ columns, and bit complexity $\ell$ in time $\tilde{O}(n \cdot \ell)$ such that

$$\left\|\mathbf{H}-\mathbf{B}\mathbf{B}^*\right\|_{\operatorname{F}}<\frac{1}{2^{\ell}}.$$

Such an algorithm would give a linear-time implicit representation of a symmetric factorization for positive definite Hankel matrices, avoiding the need to output the generally large matrix $\mathbf{B}$ explicitly. The conjecture concerns the existence of this faster positive-semidefinite factorization, whereas the paper's preceding results use factorizations involving a difference $\mathbf{B}\mathbf{B}^*-\mathbf{C}\mathbf{C}^*$.