  We consider the variational problem associated with the Freidlin--Wentzell
Large Deviation Principle of the Stochastic Heat Equation (SHE). The logarithm
of the minimizer of the variational problem gives the most probable shape of
the solution of the Kardar--Parisi--Zhang equation conditioned on achieving
certain unlikely values. Taking the SHE with the delta initial condition and
conditioning the value of its solution at the origin at a later time, under
suitable scaling, we prove that the logarithm of the minimizer converges to an
explicit function as we tune the value of the conditioning to $ 0 $. Our result
confirms the physics prediction Kolokolov and Korshunov (2009), Meerson,
Katzav, and Vilenkin (2016), Kamenev, Meerson, and Sasorov (2016).
