Kellner's conjecture on the term count and Bell-polynomial values of the polynomials $\psi_\nu$

Let $\mathcal{P}(n)$ be the partition function, and define

$$\mathcal{P}_{\Sigma}(\nu)=\sum_{i=1}^{\nu}\mathcal{P}(i).$$

Let $\mathbb{B}_n$ be the complete Bell polynomial, and define the alternating harmonic number

$$\overline{H}_n=\sum_{i=1}^n\frac{(-1)^{i+1}}{i}.$$

For the polynomials $\psi_\nu$ occurring in Kellner's congruence for the Wilson quotient, Kellner's conjecture. (i) The polynomial $\psi_\nu$ has exactly $\mathcal{P}_{\Sigma}(\nu)$ terms. (ii)

$$\psi_\nu(\{\pm1\})=-\mathbb{B}_\nu(\{\mp i!\overline{H}_i\}_{i=1,\dots,\nu}),$$

and

$$\sum_{\nu=1}^{\infty}\frac{1}{\nu!}\psi_\nu(\{\pm1\})T^\nu=1-(T+1)^{\mp\frac{1}{1-T}}.$$

Here the signs are chosen correspondingly. These assertions concern the explicit structure and specializations of the $p$-independent polynomials $\psi_\nu$ that arise in higher congruences for Wilson quotients; the supplied text does not establish whether Kellner's assertions are resolved, so their database status remains open.