We have studied the photoluminescence (PL) emission from a GaAs/AlAs asymmetric-double-well superlattice with indirect band gap as a function of temperature and applied electric field. The PL spectrum consists of two peaks, with a separation of only 21 meV, originating from the recombination of wide-well heavy holes with either $X$ electrons of the barrier or $\Gamma$ electrons of the wide well, respectively. The PL intensity ratio of the two peaks shows a strong nonthermal population between $\Gamma$ and $X$ states near flatband, because in this sample the transfer from $\Gamma$ to $X$ after photoexcitation can be induced only by absorption of phonons with a large momentum. The small $\Gamma$-$X$ separation and the long recombination time of $X$ electrons results in the filling of the $X$ states that are at the same energy as the $\Gamma$ minimum, blocking $\Gamma$-$X$ transfer by phonon emission or interface (roughness) scattering.