Polynomials on parabolic manifolds

A Stein manifold $X$ is called $S$-parabolic if there exists a special exhaustion function on it
$$ \rho(z)\colon\{\rho(z)\leq C\}\Subset X\quad \forall C\in\mathbb R, \quad (dd^c\rho)^n=0\quad \text{outside of some compact $K\Subset X$}. $$

Apparently, for the first time, $S$-parabolic manifolds were used by Griffiths and King in the construction of the multidimensional Nevanlinna theory for the mapping $f\colon N\to M$ $S$ is a parabolic $n$-dimensional manifold $N$ onto a compact $m$-dimensional manifold $M$. Subsequently, parabolic manifolds were studied in the works of Stahl, Fact, Aituna, Zeriahi and other mathematicians. Note that the manifold $(\mathbb C^n,\ln|z|)$ is one of the simplest examples of parabolic manifolds.
This report is devoted to the space of polynomials on $S$-parabolic manifolds. A function $p\in\mathcal{O}(X)$ is called a $\rho$-polynomial on $(X,\rho)$ if it admits an estimate of $|p(z)|\leq c+d\rho(z)$, where $c,d$ are some constants. The report will give an example of a parabolic manifold $(X,\rho)$ on which there is no nontrivial polynomial $p(z)\neq \mathrm{const}$.
Definition. $S$ is a parabolic manifold $(X,\rho)$ is called regular if the polynomial space $\mathcal{P}(X)$ is dense in $\mathcal{O}(X)$. Note that the manifold $(\mathbb C^n,\ln|z|)$ or an algebraic variety with a suitable exhaustion function $\rho(z)$ are regular. The main purpose of the report is to prove an analogue of the classical Bernstein–Walsh theorem for a regular parabolic manifold relating the velocity of the polynomial approximation of the function $f(z)\in C(K)$ with an analytic continuation of $f$ to some neighborhood of $G\supset K$. The domain $G$ is defined by the Green function of the compact $K.$ For $(\mathbb C^n,\ln|z|)$ this analog is proved by J. Sichak.