Positivity conjecture for fractional Gaussian noise predictor coefficients

Let $H>1/2$, and consider fractional Gaussian noise with Hurst parameter $H$. For any element of this process, project it onto any finite number of its subsequent elements, and denote the resulting projection coefficients by $\Gamma_n^k$, with $2\le k\le n$. Positivity conjecture. If $H>1/2$, then all coefficients of these projections are strictly positive. This conjecture concerns the positivity underlying monotonicity properties of the Cholesky decomposition of covariance matrices for fractional Gaussian noise and fractional Brownian motion. Numerical evidence supports it, but the analytic proof remains open, even for certain low-dimensional cases.