  We consider a large number of Bosons with interaction potential $v_N(x)=N^{3
\beta}v(N^{\beta}x)$. In earlier papers we considered a set of equations for
the condensate $\phi$ and pair excitation function $k$ and proved that they
provide a Fock space approximation to the exact evolution of the condensate for
$\beta <\frac{1}{3}$. This result was extended to the case $\beta<\frac{1}{2}$
by E. Kuz, where it was also argued informally that the equations of our
earlier work do not provide an approximation for $\beta>\frac{1}{2}$. In 2013,
we introduced a coupled refinement of our original equations and conjectured
that they provide a Fock space approximation in the range $0 \le \beta < 1$. In
the current paper we prove that this is indeed the case for $\beta <
\frac{2}{3}$, at least locally in time. In order to do that, we re-formulate
the equations of \cite{GMM} in a way reminiscent of BBGKY and apply harmonic
analysis techniques in the spirit of X. Chen and J. Holmer to prove the
necessary estimates. In turn, these estimates provide bounds for the pair
excitation function $k$.
