That brief IS the Administration position. It's the US DOT brief in the case.

The issue they're fighting over has nothing, really, to do with HSR. It has to do with who decides what the criteria and standards are for measuring passenger service performance by the railroads related to Amtrak trains. Basic argument by the RRs is that Amtrak should have no part in the process because it's not "the government" - a position that Congress made possible by saying Amtrak is to be managed as if they are "for profit" while laying all sorts of operating requirements on them and treating them as a de facto government agency.

Given the current makeup of the Supreme Court, I would expect them to look for any way they can hand a decision to the railroads against Amtrak. However, the law they're fighting over requires consultation with all and sundry including the railroads, so even if they throw out Amtrak as a co-decisionmaker regarding the measurement standards, Amtrak should still be able to participate in the process - would be really uncool for the Court to lock them out completely.

Interestingly, the poster child for bad Amtrak timekeeping in the past (Sunset) seems to be working quite well these days looking randomly at the train tracker on Amtrak's web site. In fact, the only mostly-UP-hosted long distance train that routinely gets delayed is the Zephyr. OTOH, the old days (not so long ago) of BNSF treating Amtrak right seem to be gone, completely. Hard to find a mostly-BNSF-hosted train (other than, sometimes, the Chief) that runs even close to time, anywhere. So the underlying issue is still there.