Kronecker congruence conjecture for Ramanujan's continued fraction of order 10

Let $U(\tau)$ be the modular function considered above, and for an odd prime $p\neq 5$ let $U_p(X,Y)=0$ denote its modular equation of level $p$. Kronecker congruence conjecture. For an odd prime $p\neq 5$, the modular equation satisfies

$$U_p(X,Y)\equiv\begin{cases} (X^p-Y)(X-Y^p)\pmod{p\mathbb{Z}[X,Y]}, &\text{if }p\equiv\pm 1\pmod{10},\\ (X^p-Y)(X+Y^p)\pmod{p\mathbb{Z}[X,Y]}, &\text{if }p\equiv\pm 3\pmod{10}. \end{cases}$$

This conjecture is proposed from the modular equations computed for $U(\tau)$, whose associated modular curve has genus one and therefore is not accessible by the methods used for the analogous modular equations of $g(\tau)$.