Minimal-energy conjecture for endomorphisms of noncommutative tori

Let $A_\theta$ be a noncommutative torus with smooth subalgebra $A_\theta^\infty$, and let $\varphi_A$ be the $*$-automorphism defined by

$$\varphi_A(u)=u^pv^q,\qquad \varphi_A(v)=u^rv^s,$$

where A=\begin{pmatrix}p&q\r&s\end{pmatrix}\in SL(2,\mathbb Z). For a $*$-endomorphism $\varphi\colon A_\theta^\infty\circlearrowleft$, let $\mathcal{L}(\varphi)$ be the action functional defined using the canonical derivations and trace, and suppose that $\varphi$ induces $A$ on $K_1(A_\theta)\cong\mathbb Z^2$. Minimal-energy conjecture. The value

$$\mathcal{L}(\varphi_A)=4\pi^2(p^2+q^2+r^2+s^2)$$

is minimal among all such $\mathcal{L}(\varphi)$. This asserts that the natural linear $*$-automorphism realizes the minimum action in each prescribed $K$-theory class. No resolution status is supplied in the source, so the conjecture is recorded as open.