This Washington Post editorial argues that they should.
It first notes the justification for not disclosing reasons for recusals:
Justices have traditionally declined to elaborate on why they’ve stepped aside. One reason: a legitimate concern that revealing the cause for a recusal could empower future litigants to manufacture conflicts — such as hiring the spouse or child of a justice as a lawyer on the case — to force the removal of a justice who appears philosophically hostile to their arguments.
It then argues that the most recent recusals of Justice Roberts and Justice Breyer from a pending securities suit were likely due to their ownership of stock in the parent corporation of a party. Justice Roberts is since back on the case, likely because he sold his stock.
The editorial argues that since such financial stakes are eventually revealed in the justices’ annual financial disclosure statements, there is no harm in revealing stock ownership as a reason for recusal. But it never explains how that translates into an argument for disclosing the reasons for all recusals.
Its more convincing point is that the lawyers in the case can generally figure out the reason for the recusal anyway, so why keep the public in the dark?
Thanks to ABA Journal for the link.