we use a joint likelihood of the form $$\mathcal{L}(C_\mathrm{O1},C_\mathrm{O2} | \Theta, \mathcal{H}) = \mathcal{L}(C_\mathrm{O1} | \Theta, \mathcal{H}) \mathcal{L}(C_\mathrm{O2} | \Theta, \mathcal{H}).$$

As discussed above, we adopt a power-law model for the energy-density spectrum of the stochastic background, giving the model spectrum $\langle C (f) \rangle = \Omega_\mathrm{ref}\left(f/f_\mathrm{ref}\right)^\alpha$. We consider different two prior probability density distributions on $\Omega_\mathrm{ref}$: a uniform distribution $p(\Omega_{\rm ref}) \propto 1$ and a log-uniform distribution $p(\Omega_{\rm ref})\propto \Omega_{\rm ref}^{-1}$. We use a peaked prior on the spectral index: $p(\alpha) \propto 1-\alpha/|\alpha_\mathrm{max}|$ with $\alpha_\mathrm{max} = 8$, preferring shallow slopes while still allowing for spectral indices far steeper than those predicted for known sources.

When allowing for alternative gravitational-wave polarizations, we will treat the total canonical energy density as a sum of three distinct power laws for the tensor, vector, and scalar-polarized components of the stochastic background. In this case, our model cross-correlation spectrum becomes (see Eq. ) $$\begin{aligned}
\langle C(f)\rangle_\mathrm{TVS} = \Omega^T_\mathrm{ref}(&f/f_\mathrm{ref})^{\alpha_T} + \beta_V(f)\Omega^V_\mathrm{ref}(f/f_\mathrm{ref})^{\alpha_V} \\
&+ \beta_S(f)\Omega^S_\mathrm{ref}(f/f_\mathrm{ref})^{\alpha_S},
\end{aligned}$$ where $\Omega^A_\mathrm{ref}$ and $\alpha_A$ are the amplitudes and spectral indices of each polarization component. Note that $\beta_T(f)=\gamma_T(f)/\gamma_T(f)=1$. Table \[tab:nongrUL\] lists the 95% credible upper limits on the amplitudes $\Omega^A_\mathrm{ref}$, following marginalization over spectral indices.

Detailed Parameter Estimation Results
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A. 