Lagrange interpolation at Chebyshev nodes: realise every closed set as its limit points (Erdős #1151)

Given nodes $a_1,\ldots,a_n\in[-1,1]$, let $$\mathcal{L}^n f(x)=\sum_{1\le i\le n} f(a_i)\,\ell_i(x)$$ be the Lagrange interpolation polynomial of degree $n-1$ that agrees with $f$ at the $a_i$ (the $\ell_i$ are the fundamental Lagrange polynomials for these nodes). Take the $a_i$ to be the Chebyshev nodes. Prove that for any closed set $A\subseteq[-1,1]$ there exists a continuous function $f:[-1,1]\to\mathbb{R}$ such that $A$ is exactly the set of limit points of the sequence $\big(\mathcal{L}^n f(x)\big)_{n}$ (as $n\to\infty$ over the Chebyshev interpolation schemes). As recorded on the source page, the precise quantifier on the evaluation point $x$ is left ambiguous by the original formulation; the concrete version that Erdős studied fixes a point of the form $x=\cos(\pi p/q)$ with odd integers $p,q\ge 1$.

Acceptance. FULLY RESOLVES: a complete proof, at the appropriate evaluation point $x=\cos(\pi p/q)$ (odd $p,q$), that every closed $A\subseteq[-1,1]$ is the set of limit points of $\big(\mathcal{L}^n f(x)\big)$ for some continuous $f$ — giving the construction of $f$ from $A$ — OR a proof for the arbitrary/fixed-$x$ reading once the quantifier is disambiguated; machine-checkable (Lean/Coq) preferred, otherwise a full written proof. ADVANCES, any of: (a) realise the case $A=\{a\}$ a single point (a genuine strengthening of Erdős's [Er41] divergence result, which only realises the $+\infty$ behaviour); (b) realise every finite closed $A$; (c) rigorously resolve the quantifier ambiguity by proving or disproving one of the natural readings of the Va99 statement; (d) extend Erdős's [Er41] divergence construction to a strictly larger class of evaluation points $x$. Each milestone requires the explicit construction plus a complete proof (or Lean artifact). Deliver the construction and proof, or a certified counterexample to a stated reading.