  We present upper limits on the 21 cm power spectrum at $z = 5.9$ calculated
from the model-independent limit on the neutral fraction of the intergalactic
medium of $x_{\rm H{\small I }} < 0.06 + 0.05\ (1\sigma)$ derived from dark
pixel statistics of quasar absorption spectra. Using 21CMMC, a Markov chain
Monte Carlo Epoch of Reionization analysis code, we explore the probability
distribution of 21 cm power spectra consistent with this constraint on the
neutral fraction. We present 99 per cent confidence upper limits of
$\Delta^2(k) < 10$ to $20\ {\rm mK}^2$ over a range of $k$ from 0.5 to $2.0\
h{\rm Mpc}^{-1}$, with the exact limit dependent on the sampled $k$ mode. This
limit can be used as a null test for 21 cm experiments: a detection of power at
$z=5.9$ in excess of this value is highly suggestive of residual foreground
contamination or other systematic errors affecting the analysis.
