Sugawara's conjecture on ray class invariants for split 2

Let $K$) be an imaginary quadratic field with discriminant $d_K$, let $(2)=\mathfrak p_1\mathfrak p_2$, let $h(d_K)$ denote the class number, and let $\textsf{K}_1$ denote the relevant ray class field. Sugawara's conjecture. If $d_K\equiv 1\pmod 8$, then the invariants for $\mathfrak m=\mathfrak p_1,\mathfrak p_2$ have degree $2h(d_K)$ over $\mathbb Q$, each generating $\textsf{K}_1$ over $\mathbb Q$; the invariant for $\mathfrak p_1\mathfrak p_2$ has degree $h(d_K)$ over $\mathbb Q$ and generates $\textsf{K}_1$ over $K$. The conjecture concerns the degrees and generation of ray class invariants in the split case; the surrounding argument establishes the corresponding cases for certain other factorizations of $(2)$, while this split case is proposed here.