  This article is the second of four that completely characterize a solution
space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial
differential equations (PDEs) in 2N variables that arises in conformal field
theory (CFT) and multiple Schramm-Lowner evolution (SLE). The system comprises
$2N$ null-state equations and three conformal Ward identities which govern CFT
correlation functions of $2N$ one-leg boundary operators. In the first article
(part I), we use methods of analysis and linear algebra to prove that
$\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. The analysis
of that article is complete except for the proof of a lemma that it invokes.
This article provides the proof.
  The lemma states that if every interval among $(x_2,x_3),$ $(x_3,x_4), ...
,(x_{2N-1},x_{2N})$ is a two-leg interval of $F\in\mathcal{S}_N$, then $F$
vanishes. Proving this lemma by contradiction, we show that the existence of
such a nonzero function implies the existence of a non-vanishing CFT two-point
function involving primary operators with different conformal weights, an
impossibility. This proof (which is rigorous in spite of our occasional
reference to CFT) involves two different types of estimates, those that give
the asymptotic behavior of F as the length of one interval vanishes, and those
that give this behavior as the lengths of two intervals vanish simultaneously.
We derive these estimates by using Green functions to rewrite certain
null-state PDEs as integral equations, combining other null-state PDEs to
obtain Schauder interior estimates, and then repeatedly integrating the
integral equations with these estimates until we obtain optimal bounds. In the
case where two adjacent interval lengths vanish, we use a Green function that
contains the Jacobi heat kernel as its essential ingredient.
