  We discuss a Gaussian multiplicative chaos (GMC) structure underlying a
family of random measures $\mathbf{M}_r$, indexed by $r\in\mathbb{R}$, on a
space $\Gamma$ of directed pathways crossing a diamond fractal with Hausdorff
dimension two. The laws of these random continuum path measures arise in a
critical weak-disorder limiting regime for discrete directed polymers on
disordered hierarchical graphs. For the analogous subcritical continuum polymer
model in which the diamond fractal has Hausdorff dimension less than two, the
random path measures can be constructed as subcritical GMCs through couplings
to a spatial Gaussian white noise. This construction fails in the critical
dimension two where, formally, an infinite coupling strength to the
environmental noise would be required to generate the disorder. We prove,
however, that there is a conditional GMC interrelationship between the random
measures $(\mathbf{M}_r)_{r\in \mathbb{R}}$ such that the law of $\mathbf{M}_r$
can be constructed as a subcritical GMC with random reference measure
$\mathbf{M}_R$ for any choice of $R\in (-\infty, r)$. A similar GMC structure
plausibly would hold for a critical continuum (2+1)-dimensional directed
polymer model.
