Conjecture on the radical characterization set for Bernoulli denominators

Define

$$\overline{\mathcal{R}}=\{n\geq1:\mathbb{D}_n=\operatorname{rad}(n+1)\},\qquad \mathcal{R}=\{3,5,8,9,11,27,29,35,59\}.$$

Radical characterization conjecture. One has

$$\overline{\mathcal{R}}=\mathcal{R}.$$

The surrounding results show that $\overline{\mathcal{R}}$ is finite and constrain its elements; the conjecture remains unresolved in the supplied text.