Congruences for the fourth odd-rank moment

Let $\eta_{4}(n)$ denote the fourth symmetrized moment of the odd rank. Conjecture. For every integer $n\geq 0$, the congruences

$$\eta_{4}(125n+99)\equiv 0\pmod{25},$$

and

$$\eta_{4}(625n+224)\equiv 0\pmod{125}$$

hold. These congruences are presented as numerical-evidence-based open problems in the paper; no proof or disproof is given there.