  We consider a standard binary branching Brownian motion on the real line. It
is known that the maximal position $M_t$ among all particles alive at time $t$,
shifted by $m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}} \log t$ converges in law to
a randomly shifted Gumbel variable. Derrida and Shi (2017) conjectured the
precise asymptotic behaviour of the corresponding lower deviation probability
$\mathbb{P}(M_t \leq \sqrt{2}\alpha t)$ for $\alpha < 1$. We verify their
conjecture, and describe the law of the branching Brownian motion conditioned
on having a small maximum.
