  We consider the problem of $1$-sided device-independent self-testing of any
pure entangled two-qubit state based on steering inequalities which certify the
presence of quantum steering. In particular, we note that in the $2-2-2$
steering scenario (involving $2$ parties, $2$ measurement settings per party,
$2$ outcomes per measurement setting), the maximal violation of a fine-grained
steering inequality can be used to witness certain extremal steerable
correlations, which certify all pure two-qubit entangled states. We demonstrate
that the violation of analogous CHSH inequality of steering or nonvanishing
value of a quantity constructed using a correlation function called mutual
predictability together with the maximal violation of fine-grained steering
inequality can be used to self-test any pure entangled two-qubit state in a
$1$-sided device-independent way.
