Inner-corner obstruction conjecture for four-dimensional polydisks

Let $\beta$ be a polydisk parameter, and let $\hat{\mathbf{E}}_{k+1}$ and $\mathbf{E}_{k+1}$ be the inner and outer obstruction classes defined in the paper. Their obstruction graphs intersect at

$$\hat I_{k+1}=(\hat z^{\mathrm{in}}_{k+1},\hat\lambda^{\mathrm{in}}_{k+1}).$$

Here $c_\beta$ is the polydisk embedding function.

Inner-corner obstruction conjecture. At these intersections,

$$c_\beta(\hat z^{\mathrm{in}}_{k+1})\leq\hat\lambda^{\mathrm{in}}_{k+1}.$$

Together with the established complementary lower and upper bounds, this conjecture would fully compute $c_\beta$ on the interval $[1,\mathrm{accum}(\beta)]$. The source notes that complications remain in a potential proof.