The Hurewicz-set translate conjecture in ${\mathbb R}^{\mathbb N}$

Let $H\subset {\mathbb R}^{\mathbb N}$ be a Hurewicz set, and let $C\subset {\mathbb R}^{\mathbb N}$ be a countable dimensional set. In the topological group $({\mathbb R}^{\mathbb N},+)$, write $H+C=\{h+c:h\in H,\ c\in C\}$.

Hurewicz-set translate conjecture. If $H\subset {\mathbb R}^{\mathbb N}$ is a Hurewicz set, then for each countable dimensional set $C\subset {\mathbb R}^{\mathbb N}$,

$$H+C\neq {\mathbb R}^{\mathbb N}.$$

This is proposed as an analogue for Hurewicz sets of the Galvin–Mycielski–Solovay theorem for Lusin sets and Pawlikowski's theorem for Sierpiński sets. The source presents it as an expectation, and no resolution is given.