We present two conjectures regarding the running time of computing symmetric factorizations for a Hankel matrix $\mathbf{H}$ and its inverse $\mathbf{H}^{-1}$ as $\mathbf{B}\mathbf{B}^*$ under fixed-point arithmetic. If solved, these would result in a faster-than-matrix-multiplication algorithm for solving sparse poly-conditioned linear programming problems, a fundamental problem in optimization and theoretical computer science. To justify our proposed conjectures and running times, we show weaker results of computing decompositions of the form $\mathbf{B}\mathbf{B}^* - \mathbf{C}\mathbf{C}^*$ for Hankel matrices and their inverses with the same running time. In addition, to promote our conjectures further, we discuss the connections of Hankel matrices and their symmetric factorizations to sum-of-squares (SoS) decompositions of single-variable polynomials.