  A conjecture proposed by J. Tripp in 2002 states that the crossing number of
any knot coincides with the canonical genus of its Whitehead double. In the
meantime, it has been established that this conjecture is true for a large
class of alternating knots including $(2, n)$ torus knots, $2$-bridge knots,
algebraic alternating knots, and alternating pretzel knots. In this paper, we
prove that the conjecture is not true for any alternating $3$-braid knot which
is the connected sum of two torus knots of type $(2, m)$ and $(2, n)$. This
results in a new modified conjecture that the crossing number of any prime knot
coincides with the canonical genus of its Whitehead double. We also give a new
large class of prime alternating knots satisfying the conjecture, including all
prime alternating $3$-braid knots.
