Sugawara's conjecture on ramified prime-5 conjugates

Let $K$ be an imaginary quadratic field with discriminant $d_K$, suppose $(5)=\wp_5^2$, let $\Sigma=\textsf{K}_1$, and let $\rho\in\Sigma$. Let $\psi\in\operatorname{Gal}(\Sigma/K)$. Sugawara's conjecture. If $5\mid d_K$, $(5)=\wp_5^2$, and $1\ne\psi\in\operatorname{Gal}(\Sigma/K)$, then the implication

$$\rho^\psi=a\rho\quad\Longrightarrow\quad a\not\equiv1\pmod{\wp_5}$$

holds in $\Sigma=\textsf{K}_1$. The claim concerns nontrivial Galois conjugation in the ramified prime-5 case; the paper notes that in this situation $a=\varepsilon^2$ is a square in $\Sigma$, but gives no resolution of the conjecture.