Finiteness and completeness conjecture for Bernoulli polynomial derivatives

The Bernoulli polynomials $\mathbf{B}_n(x)$ are defined by

$$\frac{t e^{xt}}{e^t-1}=\sum_{n=0}^{\infty}\mathbf{B}_n(x)\frac{t^n}{n!},$$

and let

$$\mathcal{S}=\{1,2,4,6,10,12,28,30,36,60\},\qquad \overline{\mathcal{S}}=\{n\geq1:\mathbf{B}'_n(x)\in\mathbb{Z}[x]\}.$$

Finiteness and completeness conjecture. The set $\overline{\mathcal{S}}$ is finite and

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

This conjecture asserts that the known sequence of indices whose Bernoulli polynomial derivative has integral coefficients is finite and completely determined by the displayed set; the paper gives no resolution of these assertions.