Self-conjugate-grid pointwise interpolation-error conjecture

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 self-conjugate, meaning that nonreal nodes occur with their conjugates and every real node occurs with even multiplicity. 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 pointwise conjecture. For every $0\leqslant k\leqslant n-1$,

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

If additionally $\operatorname{\mathsf{Re}}(z_{1:n})\in\mathbb{R}^n_+$, the same equality holds at $z=0$. This is a proposed extension of the pointwise worst-case interpolation result from real nodes to self-conjugate grids; its resolution is not supplied in the source.