  We show that for any $\varepsilon>0$ there is an XOR game $G=G(\varepsilon)$
with $\Theta(\varepsilon^{-1/5})$ inputs for one player and
$\Theta(\varepsilon^{-2/5})$ inputs for the other player such that
$\Omega(\varepsilon^{-1/5})$ ebits are required for any strategy achieving bias
that is at least a multiplicative factor $(1-\varepsilon)$ from optimal. This
gives an exponential improvement in both the number of inputs or outputs and
the noise tolerance of any previously-known self-test for highly entangled
states. Up to the exponent $-1/5$ the scaling of our bound with $\varepsilon$
is tight: for any XOR game there is an $\varepsilon$-optimal strategy using
$\lceil \varepsilon^{-1} \rceil$ ebits, irrespective of the number of questions
in the game.
