Critical-corner classification conjecture for skew Howe limit shapes

Let $t_+$ be the right endpoint of the support of the limit shape, let $z_+$ be the corresponding root, and let $Omega$ denote the limit shape. Critical-corner classification conjecture. The following are equivalent:

$$t_+ = c;$$ $$z_+ = 0;$$ $$\Omega'(t_+) = 0;$$ $$\int_0^1 f(s)\,\mathrm{d}s = c\int_0^1 \frac{\mathrm{d}s}{g(s)}.$$

This characterizes the critical case in which the support reaches the right boundary and the limit shape approaches the corner flatly. By symmetry, an analogous equivalence is expected at the left boundary $t_-=-1$.