The prediction said that during the 2021–2022 (117th) Congress, Senate Democratic leadership would not bring forward and pass legislation to codify Obergefell / provide federal protection for same-sex marriage before that Congress ended in early January 2023.

What actually happened:

• In response to concerns raised after the Dobbs decision and Justice Thomas’s concurrence, Congress advanced the Respect for Marriage Act (RFMA), H.R. 8404. The House first passed it on July 19, 2022. (en.wikipedia.org)
• The Senate then took it up and passed the bill on November 29, 2022, by a 61–36 vote, with 12 Republicans joining all Democrats, during the 117th Congress. (senate.gov)
• The House agreed to the Senate-amended bill on December 8, 2022, and President Biden signed it into law on December 13, 2022, still within the 117th Congress. (en.wikipedia.org)

Substance of the law:

• RFMA repeals the Defense of Marriage Act and requires the federal government and all states to recognize any marriage that is valid in the state where it was performed, explicitly covering same-sex and interracial marriages. This is widely described as "enshrining" or "protecting" marriage equality in federal law. (congress.gov)
• Legal and news analyses note that RFMA does not fully codify Obergefell in the sense of forcing every state to issue same-sex marriage licenses if the Supreme Court overturned Obergefell, but it does provide durable federal recognition and interstate recognition protections for same-sex marriages. (time.com)

How this maps to the prediction:

• The core falsifiable claim was that Democrats would not bring up and pass such legislation in the Senate while they had the chance in the 117th Congress. In reality, Senate Democratic leadership did exactly that by scheduling and passing the Respect for Marriage Act in November 2022, in coalition with sufficient Republican votes. (senate.gov)
• Even if one argues RFMA is not a perfect one-to-one codification of the Obergefell decision, it clearly is federal legislation whose central purpose is to protect same-sex marriage recognition nationwide and to backstop Obergefell at the statutory level. That directly contradicts the prediction that Democrats would decline to bring forward and pass such a bill when they had the opportunity.

Because the Senate did in fact bring forward and pass legislation providing federal protection for same-sex marriages in the 117th Congress, the prediction is wrong.