This scenario appears less likely following the extensive review conducted by the SH0ES collaboration, where several potential sources of systematics have been examined [1, 6, 7].

In Ref. [17], seven hints were proposed, suggesting that a compelling definitive solution might entail combining early and late-time new physics.

Recent BAO measurements released by the DESI Collaboration [25, 26, 27] seem to suggest dynamical dark energy, potentially reopening the avenue for new physical mechanisms at late times that could address the Hubble tension [28]. See also Refs. [29, 30, 31, 32, 33, 34, 35, 36] for discussion.

It is important to highlight that BAO measurements are sensitive to the sound horizon evaluated at the baryon drag epoch, commonly denoted by $r_{\mathrm{d}}$ [45]. Conversely, the scale pertinent to the acoustic peaks in the CMB is the sound horizon evaluated at recombination, typically denoted by $r_s(z_{*})$ [46]. These two epochs are separated in redshift by $\mathrm{\Delta }z=z_{\mathrm{d}}-z_{*}\sim 30$.

This relation is quite general and is valid for any cosmological model where spacetime is described by a pseudo-Riemannian manifold, photons propagate along null geodesics, and their number is conserved over time. However, an important caveat is that, in certain modified gravity theories, the distance inferred from standard sirens can differ from the (electromagnetic) luminosity distance. For instance, this discrepancy can arise in models featuring a running of an effective Planck mass, which rescales the luminosity due to modified friction in the GW propagation; see, e.g., Refs. [48, 49].

ET will reach $z_{\mathrm{GW}}^{\max }\approx 2.5$, while LISA will cover redshifts up to $z_{\mathrm{GW}}^{\max }\approx 8$, see Supplemental Material [52] for details.

DESI is covering redshifts $z_{\mathrm{BAO}}^{\min }\approx 0.15$ to $z_{\mathrm{BAO}}^{\max }\approx 1.85$, while Euclid is anticipated to collect data from $z_{\mathrm{BAO}}^{\min }\approx 0.65$ to $z_{\mathrm{BAO}}^{\max }\approx 2.05$.

It is worth noting that GW data points are anticipated to outnumber BAO data points significantly. Therefore, we opt to reconstruct $D_{\mathrm{L}}^{\mathrm{GW}}(z)$ and estimate it at the same redshift as BAO measurements to enhance ML performance.

We recall that to resolve the Hubble tension, early time new physics should operate to decrease the value of the sound horizon by approximately $\mathrm{\Delta }{r}_{\mathrm{d}}/r_{\mathrm{d}}\sim 5–6\%$ [15, 28, 88].

Note that we could horizontally shift the 2D blue contours toward the left of the $x$ axis in Fig. 1 while maintaining good agreement with all the three requirements we impose. This situation will imply a larger reduction in the sound horizon, increasing our ability to detect the change. In this sense, we work under reasonably conservative conditions.