Supersymmetric nonlinear sigma models arise in the theory of disordered systems and are expected to share key features with O(N)-type models. They also reveal surprising connections with probabilistic models such as the vertex-reinforced jump process and the arboreal gas, making them a rich testing ground linking supersymmetric field theory and probability. In my talk, I will consider a family of nonlinear sigma models on $\mathbb{Z}^{d}$ whose target space is the hyperbolic supermanifold $H^{2∣2+2n}$, $n \in \mathbb{N}$, introduced by Crawford as an extension of Zirnbauer's $H^{2∣2}$ model for disordered systems. I will show the exponential decay of the two-point correlation function in the high-temperature regime $T \geq C n$, with $C>0$ a universal constant, for any $n \geq 1$ and in any dimension $d\geq1$. The proof is based on the reduction to a marginal fermionic theory and combines a high-temperature cluster expansion with bounds derived via Grassmann norms. Based on joint ongoing work with M. Disertori and J. Duràn Fernàndez.