Conformal-embedding conjecture for folded Dynkin-diagram CFTs

Let $G=ADE$ and let $G'$ be a folding of $G$. Write $\mathrm{CFT}(A_1,G)$ and $\mathrm{CFT}(A_1,G')$ for the two-dimensional conformal field theories associated with these Dynkin diagrams. Conformal-embedding conjecture. The CFT associated with the folded diagram can be conformally embedded in the CFT associated with the unfolded diagram:

$$\mathrm{CFT}(A_1,G')\subset \mathrm{CFT}(A_1,G).$$

The conjecture is motivated by the observed equality of central charges under folding; the supplied text gives no proof or resolution.