  In the case (4/3)<p<4, and assuming a pair of locally finite positive Borel
measures on the real line have no common point masses, we prove two conjectures
of T. Hyt\"onen and E. Vuorinen from 2018 on two weight testing theorems for
the Hilbert transform on weighted L^{p} spaces. Namely, the two weight norm
inequality holds (1) if and only if the global quadratic interval testing
conditions hold, (2) if and only if the local quadratic interval testing, the
quadratic Muckenhoupt, and the quadratic weak boundedness conditions all hold.
We also give a slight improvement of the second conjecture in this setting by
replacing the quadratic Muckenhoupt conditions with two smaller conditions.
