Exponent at which the semiclassical constant becomes optimal

Define $\gamma_c(d) = \mathrm{inf}\{\gamma : L_{\gamma,d} = L^{\mathrm{cl}}_{\gamma,d}\}$, where $L^{\mathrm{cl}}_{\gamma,d} = \Gamma(\gamma+1)/((4 \pi)^{d/2} \Gamma(\gamma+1+d/2))$; $L_{\gamma,d}/L^{\mathrm{cl}}_{\gamma,d}$ is nonincreasing in $\gamma$ (Aizenman, Lieb 1978). Known: $1 \le \gamma_c(d) \le 3/2$ for all d (lower bound Helffer, Robert 1990; upper bound Laptev, Weidl, Acta Math 2000), $\gamma_c(1) = 3/2$, and $\gamma_c(2)$ $\ge\ \mathrm{about}\ 1.165$, the exponent at which the one-bound-state and semiclassical constants cross. Determine $\gamma_c(2)$ and $\gamma_c(3)$; the 1976 conjecture $\gamma_c(3) = 1$ follows from a proof of mathph.lieb-thirring-constants.kinetic-3d.

Proofs of $L_{\gamma,d} = L^{\mathrm{cl}}_{\gamma,d}$ for $\gamma \ge \gamma_{*}$ together with explicit potentials beating $L^{\mathrm{cl}}$ for $\gamma < \gamma_{*}$, in $d = 2$ and $d = 3$.