$C(G)$ denotes the cycle set of $G$ (the set of lengths of all cycles of $G$). This question came up recently and I have been meaning to take a stab at it, but I wanted to know if there exists some well known counterexample. Even dropping 3-regularity, the statement is not at all trivial. For example, thinking that one could simply construct such a graph $G'$ by
$$ G' = \bigoplus_{n \in C(G)} \mathcal{C}_n $$
Where $\mathcal{C}_n$ denotes the cycle graph on $n$ vertices is simply wrong, since this construction introduces more cycles in $G'$ that are not in $G$. What is perhaps more interesting, is that this statement seems to imply the Erdős–Gyárfás conjecture, if paired with the main result of this paper. Could this ever possibly be true?
The Heawood graph is a counterexample.
It is a $3$-regular graph with girth $6$; the set of cycle lengths in the graph is $\{6, 8, 10, 12, 14\}$.
A $3$-regular planar graph cannot have girth $6$. This is a bound via Euler's formula: in general, if the girth of an $n$-vertex planar graph is $k$, it has at most $\frac{k}{k-2}(n-2)$ edges, which is at most $\frac64(n-2) = \frac32n - 3$ edges in our case. But a $3$-regular $n$-vertex graph must have exactly $\frac32n$ edges.