  We consider a {\it{Gaussian multiplicative chaos}} (GMC) measure on the
classical Wiener space driven by a smoothened (Gaussian) space-time white
noise. For $d\geq 3$ it was shown in \cite{MSZ16} that for small noise
intensity, the total mass of the GMC converges to a strictly positive random
variable, while larger disorder strength (i.e., low temperature) forces the
total mass to lose uniform integrability, eventually producing a vanishing
limit. Inspired by strong localization phenomena for log-correlated Gaussian
fields and Gaussian multiplicative chaos in the finite dimensional Euclidean
spaces (\cite{MRV16,BL18}), and related results for discrete directed polymers
(\cite{V07,BC16}), we study the endpoint distribution of a Brownian path under
the {\it{renormalized}} GMC measure in this setting. We show that in the low
temperature regime, the energy landscape of the system freezes and enters the
so called {\it{glassy phase}} as the entire mass of the Ces\`aro average of the
endpoint GMC distribution stays localized in few spatial islands, forcing the
endpoint GMC to be {\it{asymptotically purely atomic}} (\cite{V07}). The method
of our proof is based on the translation-invariant compactification introduced
in \cite{MV14} and a fixed point approach related to the cavity method from
spin glasses recently used in \cite{BC16} in the context of the directed
polymer model in the lattice.
