The cholesky decomposition $S_n S_n^T$ is updated using a rank one up/downdate using the following
https://github.com/TuringLang/AdvancedMH.jl/blob/56062a892416d20a0e67c8939c88e82e2ec9b539/src/RobustAdaptiveMetropolis.jl#L163C1-L163C55
The way it's written, we have
$$\Delta S = \eta \vert\Delta\alpha\vert \frac{S U}{\Vert U\Vert}$$
and the rank one update ($\text{sgn}(\Delta\alpha) = 1$) or downdate ($\text{sgn}(\Delta\alpha) = -1$)
$$S_n S_n^T = S_{n-1} S_{n-1}^T + \text{sgn}(\Delta\alpha) (\Delta S) (\Delta S)^T$$
$$ = S_{n-1} S_{n-1}^T \pm \eta^2 (\alpha - \alpha^*)^2 S_{n-1} \frac{U U^T}{\Vert U\Vert^2} S_{n-1}^T$$
$$ = S_{n-1}\left(I \pm \eta^2 \vert\alpha - \alpha^*\vert^2\frac{U U^T}{\Vert U\Vert^2}\right)S_{n-1}^T$$
This update in the original article Vihola 2012 appears in equation 1 as
$$S_n S_n^T = S_{n-1}\left(I + \eta (\alpha - \alpha^*) \frac{U U^T}{\Vert U\Vert^2}\right) S_{n-1}^T$$
So I'm wondering if there's an extra factor of $\eta\Delta\alpha$ and if $\Delta S$ should be instead set to
$$\Delta S = \sqrt{\eta \vert\Delta\alpha\vert} \frac{S U}{\Vert U\Vert}$$
