Linear-time inverse positive definite Hankel factorization conjecture

Let $\mathbf{H} \in \mathbb{R}^{n\times n}$ be a positive definite Hankel matrix with bit complexity $\ell$ and condition number bounded by $2^\ell$. An inverse 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 $\tilde{O}(n^{\omega/2} \cdot \ell)$ such that

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

This conjecture would provide a positive-semidefinite symmetric factorization for the inverse of a positive definite Hankel matrix within the stated subquadratic running time. It complements the paper's difference-of-positive-factorizations approach for Hankel matrices and their inverses; the supplied text gives no resolution of this conjecture.