Average-case hardness of approximating probabilities for FLO circuits

Let $V$ be a passive or active fermionic linear-optics circuit initialized in the state $\left|\Psi_{\mathrm{in}}\right>$, let $\mathbf{x}_0$ be a fiducial outcome, and let $p_{\mathbf{x}_0}(V,\Psi_{\mathrm{in}})$ denote its output probability. Let $\nu$ be the Haar distribution, with $(C,\nu)=(C_{\mathrm{pas}},\nu_{\mathrm{pas}})$ for passive FLO circuits and $(C,\nu)=(C_{\mathrm{act}},\nu_{\mathrm{act}})$ for active FLO circuits. Average-case hardness conjecture. Computing a $(1/4+o(1),1/(8C))$-multiplicative approximation to $p_{\mathbf{x}_0}(V,\Psi_{\mathrm{in}})$ for a $1/(8C)$ fraction of circuits $V$ sampled from $\nu$ is $\#\mathrm{P}$-hard. This conjecture supplies the average-case hardness assumption needed to convert the sampling-to-computation theorem into a classical hardness result for fermionic sampling; its resolution is not established in the source.