Completeness conjectures for second and third Bernoulli polynomial derivatives

For $k\geq1$, define

$$\overline{\mathcal{S}}_k=\{n\geq1:\mathbf{B}^{(k)}_n(x)\in\mathbb{Z}[x]\},$$

and let $\mathcal{S}_k$ be the corresponding computable subsets, with

$$\mathcal{S}_2=\{1\!-7,9\!-13,15,16,21,25,28\!-31,36,37,55,57,60,61,70,121,190\},$$ $$\mathcal{S}_3=\{1\!-18,20\!-22,25,26,28\!-32,35\!-38,42,50,52,55\!-58,60\!-62,66,70\!-72,78,80,92,110,121,122,156,176,177,190,191,210,392\}.$$

Second- and third-derivative completeness conjecture. One has

$$\overline{\mathcal{S}}_2=\mathcal{S}_2\qquad\text{and}\qquad\overline{\mathcal{S}}_3=\mathcal{S}_3.$$

The paper proves finiteness of all $\overline{\mathcal{S}}_k$ and records these computed sets, but does not establish their completeness.