Möbius inversion formula for class numbers in terms of the floor-function sum

Let $n$ be an odd squarefree positive integer. Define $\delta(n)$ by

$$\delta(n)=\begin{cases}1,&n\equiv3\pmod{4},\\0,&\text{otherwise}. \end{cases}$$

The Möbius-inversion class-number conjecture. One has

$$h^*(-n)\delta(n)=\sum_{d\mid n}\mu\left(\frac nd\right)\left(\frac{1-d}{2}\delta(d)-2f(d)\right).$$

The paper derives this as a conjectural consequence of the preceding odd-squarefree identity and notes that it would express all $h^*(-n)$ for odd squarefree $n>0$ in terms of the sum defining $f$; no general proof is supplied.