Divisibility of the class number of the quadratic field k' by 16

Let $p$ be a prime, let $k'$ be the quadratic field and $\varepsilon'$ a unit of $k'$ occurring in the algorithm, and let $h_{k'}$ denote its class number. Let $\eta_1$ and $\eta_2$ be the quantities defined by the algorithm, and let $\gamma$ denote its associated automorphism. Class-number divisibility conjecture. Assume $p\equiv1\pmod{16}$ and $N_{k'/\mathbb{Q}}(\varepsilon')=-1$. Then $h_{k'}$ is divisible by $16$ if and only if $\eta_1\eta_2^{1+\gamma}>0$. Consequently, if $(-1)^{(p-1)/16}\eta_2^{1+\gamma}<0$, then $a_1=a_2$. This conjecture links the sign condition arising in the algorithm to divisibility of the class number by $16$; the supplied text does not indicate whether it has been proved or disproved.