Constant-term spectral-action conjecture for noncommutative tori

Let $\mathcal{D}_A$ be the fluctuated Dirac operator on a noncommutative $n$-torus, and let $\mathcal{D}+A$ denote the corresponding operator on the commutative $n$-torus. Consider the constant terms in their spectral actions.

Constant-term spectral-action conjecture. The constant term of the spectral action of $\mathcal{D}_A$ on the noncommutative $n$-torus is proportional to the constant term of the spectral action of $\mathcal{D}+A$ on the commutative $n$-torus.

This conjecture compares the dimension-independent constant contribution of the spectral action in the noncommutative and commutative settings. The supplied text gives no resolution or precise proportionality factor, so the conjecture remains open.