Thanks to the friend who alerted me to TRIP (The Rest is Politics), episode 233, From Peerages to Putin: the corruption in global politics. If you missed it, take a listen. In it, Alastair Campbell and Rory Stewart also mention this cross-party report: https://www.ukgovernanceproject.co.uk/ as chaired by the Rt Hon Dominic Grieve

Of particular relevance to this petition (from the Summary of Recommendations) is Recommendation 5: Political Honours:

S.6. The main elements of this Recommendation are as follows: 
- a. Prime Ministers would no longer make personal recommendations for honours. All nominations (from the Prime Minister or another source) would go to the Independent Committees. The Main Committee would have ultimate powers to approve or reject. No10 would not be able to influence the outcomes of the Honours process.
- b. The State and Political Honours Committees should be merged to create a new, central and Independent Committee, with an independent Chair, which would provide greater scrutiny of nominations from political parties and the Civil Service.
- c. Political honours should go through the same process as for other nominations, requiring evidence of public service beyond the usual role
- d. The principles governing the honours system, and the criteria for a nomination to be approved, should be explicitly stated and published.