Equality of the lower and upper support functionals on the boundary of the parameter simplex

Let $T$ be a tensor over an algebraically closed field, let $\Theta$ be the parameter simplex for support functionals, and let $\Theta_{(\varkappa)}$ denote the relevant boundary edge of $\Theta$. The lower and upper support functionals are denoted by $\zeta_\theta$ and $\zeta^\theta$, respectively.

Boundary support-functional conjecture. For every $\theta\in\Theta_{(\varkappa)}$, one has

$$\zeta_\theta(T)=\zeta^\theta(T).$$

The two support functionals are known to be equal for oblique tensors, while Bürgisser showed that they are separated for generic tensors when $\theta$ lies in the relative interior of $\Theta$. This conjecture concerns the remaining boundary case, motivated by rigidity results for support functionals whose parameter lies on an edge of $\Theta$.