Parity conjecture for logarithmic singularities in the Suslov problem

Let equation have regular singularities $s_1=s_2^\star=\mathrm{i}$, with exponent differences $\Delta_1=\Delta_2=p$, where $p\in\mathbb{Z}^{\ast}$. A singular point is called logarithmic when a local solution expansion contains logarithmic terms, and non-logarithmic otherwise.

Parity conjecture. Singular points $s_1$ and $s_2$ are non-logarithmic for any odd integer $p$ and logarithmic for any even $p\in\mathbb{Z}^{\ast}$.

The claim describes the parity dependence of logarithmic terms at the two singularities whose exponent differences are integral. The paper notes that direct calculations establish the pattern for small values of $p$, while the assertion for all nonzero integers $p$ remains conjectural.