This is the abstract of the linked paper:
Supernova (SN) cosmology is based on the key assumption that the luminosity standardization process of Type Ia SNe remains invariant with progenitor age. However, direct and extensive age measurements of SN host galaxies reveal a significant ($5.5\sigma$) correlation between standardized SN magnitude and progenitor age, which is expected to introduce a serious systematic bias with redshift in SN cosmology. This systematic bias is largely uncorrected by the commonly used mass-step correction, as progenitor age and host galaxy mass evolve very differently with redshift. After correcting for this age bias as a function of redshift, the SN data set aligns more closely with the $w_0w_a$
cold dark matter (CDM) model recently suggested by the Dark Energy Spectroscopic Instrument (DESI) baryon acoustic oscillations (BAO) project from a combined analysis using only BAO and cosmic microwave background (CMB) data. This result is further supported by an evolution-free test that uses only SNe from young, coeval host galaxies across the full redshift range. When the three cosmological probes (SNe, BAO, and CMB) are combined, we find a significantly stronger ($>9\sigma$) tension with the $\Lambda$CDM model than that reported in the DESI papers, suggesting a time-varying dark energy equation of state in a currently non-accelerating universe.
And here's an attempt to make a less-technical version:
One of the key lines of evidence for dark energy is supernovae data, where faraway supernovae appear to be receding faster than they should. This line of evidence relies on the idea that supernovae brightness remains constant throughout the universe's history. This assumption might not hold; measurements suggest that there is a $5.5\sigma$1 correlation between the supernova's brightness and the age of the universe. After correcting for this bias, the supernovae data appears to agree with the recent DESI $w_0w_a$ model2 of dark matter. Combining the supernovae data with other lines of evidence (BAO/CMB), there is a $9\sigma$3 disagreement with the standard model of cosmology ($\Lambda$CDM), suggesting that dark energy is changing, and the universe's expansion is currently not accelerating.
1 This is a pretty big disagreement.
2 This is a relatively recent measurement, by the DESI collaboration, suggesting that dark energy is changing.
3 As you might guess, if $5.5\sigma$ is big, $9\sigma$ is even bigger.
In other words, if the results are robust, then dark energy is not what it's commonly treated to be (a cosmological constant). Instead, it is becoming weaker over time - potentially to the point that the universe's expansion is no longer accelerating.
There is "no real proof", of course, because all of the above hinges on if the results are robust. Scientific results are often wrong (because of some unaccounted systemic effect, confounding variable, etc.). This result is still quite new, there might be some flaw in the analysis.