  The critical $2d$ Stochastic Heat Flow (SHF) is a stochastic process of
random measures on ${\mathbb R}^2$, recently constructed in [CSZ23]. We show
that this process falls outside the class of Gaussian Multiplicative Chaos
(GMC), in the sense that it cannot be realised as the exponential of a
(generalised) Gaussian field. We achieve this by deriving strict lower bounds
on the moments of the SHF that are of independent interest.
