Aliasing estimate for ultraspherical rectangular collocation

Let $p_n(x;\lambda)$ be the degree-$n$ ultraspherical polynomial, let $P=(x_1,\ldots,x_n)$ be its roots, and write

$$\mathbf F_n(\mu_\lambda)\mathbf P_{\lambda\to P}=\begin{bmatrix}\mathbf a_1&\mathbf a_2&\cdots\end{bmatrix},$$

where $\mathbf a_j=\mathbf e_j$ for $j=1,\ldots,n$. Here $n$ and $j$ are positive integers, and $\|\cdot\|_{\ell^2}$ denotes the Euclidean norm of the column vector.

Aliasing estimate. There exists $c(\lambda)>0$ such that

$$\|\mathbf a_j\|_{\ell^2}\leq c(\lambda)$$

for all $n,j$.

The preceding proposition gives a Frobenius-norm estimate growing linearly with the number of aliased columns; this claim would provide a uniform columnwise bound instead. The supplied text does not state whether the estimate has been proved or disproved.