  We investigate the relationship between projective measurements and positive
operator-valued measures (POVMs) in the task of quantum steering. A
longstanding open problem in the field has been whether POVMs are more powerful
than projective measurements for the steerability of noisy singlet states,
which are known as Werner states. We resolve this problem for two-qubit systems
and show that the two are equally powerful, thereby closing the so-called
Werner gap. Using the incompatible criteria for noisy POVMs and the connection
between quantum steering and measurement incompatibility, we construct a local
hidden state model for Werner states with Bloch sphere radius $r\leq 1/2$ under
general POVMs. This construction also provides a local hidden variable model
for a larger range of Werner states than previously known. {In contrast, we
also show that projective measurements and POVMs can have inequivalent noise
tolerances when using a fixed state ensemble to build different local hidden
state models.} These results help clarify the relationship between projective
measurements and POVMs for the tasks of quantum steering and nonlocal
information processing.
