Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the unique interpolating polynomial $p_n(x)$ of $f$ at the nodes $\{x_0,x_1,...,x_n\}$ satisfy $p_n(x)\ge 0,\forall x\in [a,b]$ ?

compare with Does every positive continuous function have a non-negative interpolating polynomial of every degree?

In the present question, we do not want to let the nodes vary with the function.

NOTE: If there are infinitely many $n$ for which such distinct points do not exist, then that is a strong enough statement to imply Faber's theorem.

Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the unique interpolating polynomial $p_n(x)$ of $f$ at the nodes $\{x_0,x_1,...,x_n\}$ satisfy $p_n(x)\ge 0,\forall x\in [a,b]$ ?

compare with Does every positive continuous function have a non-negative interpolating polynomial of every degree?

In the present question, we do not want to let the nodes vary with the function.

Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the unique interpolating polynomial $p_n(x)$ of $f$ at the nodes $\{x_0,x_1,...,x_n\}$ satisfy $p_n(x)\ge 0,\forall x\in [a,b]$ ?

compare with Does every positive continuous function have a non-negative interpolating polynomial of every degree?

In the present question, we do not want to let the nodes vary with the function.

Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the unique interpolating polynomial $p_n(x)$ of $f$ at the nodes $\{x_0,x_1,...,x_n\}$ satisfy $p_n(x)\ge 0,\forall x\in [a,b]$ ?

compare with Does every positive continuous function have a non-negative interpolating polynomial of every degree?

In the present question, we do not want to let the nodes vary with the function.

NOTE: If there are infinitely many $n$ for which such distinct points do not exist, then that is a strong enough statement to imply Faber's theorem.