Strassen's support-functional conjecture for tight tensors

Let $T$ be a tight tensor. Its asymptotic spectrum is the collection of spectral points, namely real-valued semiring homomorphisms on the tensor semiring that are monotone under degeneration. For each probability distribution $\theta=(\theta_A,\theta_B,\theta_C)$ on $\{1,2,3\}$, let $\widehat{\zeta}^{\theta}$ be Strassen's support functional on tight tensors.

Strassen's support-functional conjecture. The asymptotic spectrum of the class of tight tensors coincides with the set of support functionals $\widehat{\zeta}^{\theta}$.

Equivalently, the support functionals should determine the entire spectrum on tight tensors. Strassen proved that the support functionals determine the asymptotic subrank of a tight tensor; the conjecture asks that they determine the full asymptotic spectrum, and it remains open.