This letter is not from a participant (ie., a name on the actual docket) in the case. I venture the court receives hundreds and thousands of such letters, so I do not expect the response will be fast tracked with all the business there.
Understand each judge is assisted by 4 attorneys (for that court, think Yale, Harvard, etc. graduates - far from stupid individuals). I bet the farm the judge (or judges), if he does not immediately have the answer, will tell that assistant to review/investigate the allegations and give him an opinion on the truth of the claim (with thousands of cases, these judges cannot simply drop the process of decisions on hundreds of cases and turn all attention to a concerned citizen). They absolutely take the allegations seriously, but I would not anticipate action immediately (it is an ethical matter separate from the case). Fans may be outraged, but these judges are not likely to act like fans and have some important cases on life, liberty and property to address as ongoing business.
Allegations are not frowned upon. By the parties, there are tactical reasons not to make the allegations (unfortunately, it can be a part of theatrics/melodrama and start to grate on the court prior to a decision in the "boy who cried wolf" sense). These ethical complaints are reviewed and referred to committees for action all the time.
Lost in all this discussion is oral argument does not add to the existing record. The assumption from the media and others is the judges somehow are incapable of reviewing the record (the only real evidence on appeal) and coming to an opinion on the subject without regard to advocate statements in oral argument. Any opinion representing a decision from that court is likely reviewed/drafted by 3 non-judge attorneys (their law clerks) and drafted/reviewed/edited and approved by the authoring judges (the final document reflects their opinion). The notion that these judges will miss something that armchair lawyers/non-lawyers have pointed out in the media is unlikely at best. Similarly, the odds of a judge basing a decision on some statement at argument that is not an "I have no claim here" concession is about zero.
For those trying to read oral argument exchanges, understand these judges deal with the smartest attorneys out there. The tone/statements of those discussions may be purely tactical in an effort to elicit a result. Many oral arguments appear to go badly, but the decision is 180 degrees out from how that argument appeared to go. I tend to put little weight on what is said, because it infrequently anticipates a result (silence can often be far worse, only worse than a denial of an oral argument request). In reality, these decisions are based on the closed universe of the cold trial record and possibly the briefs. Far less exciting than the riveting television court drama, but more often the reality of appellate practice.