Base-k representations of the co-domain of a polynomial - is it context-free? In chapter 4 of Jeffrey Shallit's A Second Course in Automata Theory the following problem is listed as open:

Let $p(n)$ be a polynomial with rational coefficients such that $p(n) \in \mathbb{N}$ for all $n \in \mathbb{N}$. Prove or disprove that the language of the base-k representations of all integers in $\{p(n) \mid n \geqslant 0\}$ is context-free if and only if the degree of $p$ is $\leqslant 1$.

What's its status now (as for Oct 2018)? Is it proven? What about some special cases?
Of course $k \geq 2$ here.

There once was a manuscript by Horváth that claimed to solve the problem, but it was unclear in several places and to my knowledge was never published.

As far as I know, the problem is still open.  One direction of the implication is easy, of course.