Sugawara's conjecture on automorphisms of a ray class field

Let $d$ be an integer with quadratic discriminant $d_k=\prod_{p\mid d}p^*$, where $p^*=(-1)^{(p-1)/2}p$ for odd $p$, and let $2^*\in\{-4,8,-8\}$. Let $\textsf{K}_{\wp_5'}$ be the indicated ray class field, let $b$ be the unit used in the construction, and let $\psi\in\operatorname{Gal}(\textsf{K}_{\wp_5'}/K)$. Sugawara's conjecture. Assume that $-d\equiv1,4\pmod {15}$ and the $2$-factor of $d_k$ is not $2^*=-4$. If $b/b^\psi\in(\textsf{K}_{\wp_5'}^\times)^2$ for some $\psi$, then $\psi=1$. Moreover, there is a unique $\psi$ for which $b/b^\psi=-\textsf{B}^2$ with $\textsf{B}\in\textsf{K}_{\wp_5'}$, namely the automorphism $\psi:b\mapsto-1/b$. This conjecture describes the square-class behavior of the unit $b$ under the relevant Galois group; the preceding argument motivates it, but no resolution is given here.