Minimum shear viscosity to entropy ratio of the unitary gas
In plain words
Viscosity divided by entropy measures how close a fluid comes to flowing with no internal friction. A conjectured lower limit is 1/(4 pi) in natural units, and the unitary gas is among the closest fluids to it.
Precise statement
Homogeneous unitary Fermi gas ($1/a = 0$). Measure $\eta/s$ as a function of $T/T_F$ through $T_c = 0.167\,T_F$, in units of $\hbar/k_B$, and locate its minimum. Answer: minimum value and position with 10% uncertainty, including the two-fluid regime below $T_c$.
What would settle it
Uniform-box measurements of viscous damping with local entropy thermometry, or a controlled Kubo-formula calculation that matches them.
Status in the literature
Trap-averaged measurements up to 2015 placed the minimum at roughly 0.4 to 0.5 $\hbar/k_B$ near $T_c$, several times $1/(4\pi)$; a 2020 uniform-box measurement found sound diffusivity of order $\hbar/m$ near $T_c$ (Patel et al., Science 2020), but a homogeneous $\eta/s(T)$ curve through $T_c$ is lacking.