Quadratic growth conjecture for the Heisenberg algebra

Let $\mathfrak{h}_n$ be the Heisenberg algebra, and let $L(R;\mathfrak{h}_n)$ denote the gradient Lipschitz constant for the objective class under consideration on the radius-$R$ region.

Quadratic growth conjecture.

$$L(R;\mathfrak{h}_n)=\Theta(R^2).$$

The Heisenberg algebra is nilpotent and has no hyperbolic elements, but its bracket structure differs from that of $\mathfrak{se}(3)$. Establishing matching quadratic upper and lower bounds would determine whether the $R^2$ growth observed for $\mathfrak{se}(3)$ persists in this setting and would clarify whether the same exponent holds more broadly for non-compact Lie algebras without hyperbolic elements.