Uniform interpolation-error conjecture for self-conjugate grids

Let $I$ be the interpolation interval, let $\mathbb{D}$ be the unit disc, let $\mathscr{E}_n(\mathbb{D})$ be the entire functions $g:\mathbb{C}\to\mathbb{C}$ satisfying $\|g^{(n)}\|_{\mathbb{D}}\leqslant1$, and let $z_{1:n}\in\mathbb{D}^n$ be a self-conjugate grid. Let $f(\cdot\mid z_{1:n})$ be the nodal polynomial and $\hat g(\cdot\mid z_{1:n})$ the interpolation polynomial.

Self-conjugate-grid uniform conjecture. For every $0\leqslant k\leqslant n-1$,

$$\operatorname{\mathsf{\sup}}_{g\in\mathscr{E}_n(\mathbb{D})}\|g^{(k)}(\cdot)-\hat g^{(k)}(\cdot\mid z_{1:n})\|_I=\frac{1}{n!}\|f^{(k)}(\cdot\mid z_{1:n})\|_I.$$

Equivalently, the source gives the corresponding supremum identity involving Schur polynomials. This would generalize the real-node result of Kallioniemi and Shadrin to self-conjugate grids in the unit disc; the source does not provide a resolution.