Conjecture for the floor-function sum and class numbers of odd squarefree integers

Let $n$ be an odd squarefree positive integer, and let $h^*(-p)=h(-p)$ for $p\geq7$ and $h^*(-3)=1/3$. The odd-squarefree class-number conjecture. If $n\equiv1\pmod{4}$, then

$$f(n)=-\frac12\sum_{\substack{d\mid n\d\equiv3\pmod{4}}}h^*(-d).$$

If $n\equiv3\pmod{4}$, then

$$f(n)=\frac{1-n}{4}-\frac12\sum_{\substack{d\mid n\d\equiv3\pmod{4}}}h^*(-d).$$

The conjecture extends several previously established special cases and is described as the opposite extreme of the prime-power results; the supplied text does not establish it in general.