Schur-polynomial formulation of the self-conjugate-grid pointwise conjecture

Let $\mathbb{D}$ be the unit disc and let $z_{1:n}\in\mathbb{D}^n$ be a self-conjugate grid. For integers $0\leq k<n\leq t$, let $\mathsf{Q}_{t,n,k}$, $\mathsf{s}_{(t-n\mid n-k-1)}$, and $\mathsf{e}_{n-k}$ denote the Schur-polynomial expressions defined in the source, with $1_n$ the all-ones vector.

Schur-polynomial pointwise conjecture. For all $0\leq k<n\leq t$,

$$|\mathsf{Q}_{t,n,k}(z_{1:n}+1_n)|\leq1.$$

If additionally $\operatorname{Re}(z_{1:n})\in\mathbb{R}^n_+$, then

$$|\mathsf{s}_{(t-n\mid n-k-1)}(z_{1:n})|\leq\binom{t}{n}|\mathsf{e}_{n-k}(z_{1:n})|.$$

This is presented as an equivalent Schur-polynomial formulation of the preceding pointwise interpolation conjecture, so it should be treated as one mathematical claim rather than an independent restatement; the source gives no resolution.