Maximally entangled quantum state - not in hilbert space?

I am reading a paper about Gaussian quantum states and the mathematical formalism used to describe them. At one point the authors say "An important example of a Gaussian state is the maximally entangled state $\Phi$. In their endnotes, they then note that

"Although this maximally entangled state does not belong to the Hilbert space, it can always be considered as a limit of a proper pure state."

Why does the maximally entangled state not belong to the Hilbert space? Is it in some sense ill-defined as a pure state?

The maximally entangled state in the Gaussian framework is the state that one gets when taking the limit of infinite energy/photon number of a particular class of state (see equation (4) of the paper you linked). From equation equation (2) it is clear that such a state is not bounded, and thus is not in $\mathcal{H}=L^2(\mathbb{R}^n)$.