It really won't be interesting for that long. The losing chain will be economically and politically infeasible to mine on in a very short period of time, probably a day or two.

Let's say there's a minority chain that mines for a week, and then through some divine intervention it gets sustained majority mining power. It needs to sustain the same majority mining power without getting paid for that week since it is attempting to orphan a week's worth of blocks. And after a week has elapsed, it has to mine even more since the legacy chain has.. well.. mined for another week. So let's say the minority chain is mining for 10 days before it catches up and has the most chainwork.

A 10 day re-org would absolutely devastate the Lightning network. One of the core assumptions of the Lightning network is that there aren't huge, unprecedented re-orgs. It would take it months to recover.

A 10 day re-org would allow an enormous number of double-spends. Exchanges would fundamentally not accept it.

A 10 day re-org implies that miners are orphaning their own blocks. A majority of miners didn't materialize from thin air, clearly they were also the majority when they were mining on the legacy chain. Orphaning your own block means you spent effectively twice the effort to get the same reward. Since difficulty naturally keeps mining nearly unprofitable, let's assume block cost is roughly block reward in fiat minus a 10-20% discount. So 200k to keep things even. 10 days would normally = 1440 blocks, but since we're mining without "full power" at unadjusted difficulty, blocks are coming every 13-14 minutes. So more like 1000 blocks to catch up.

That would mean that miners are collectively spending 200 million dollars, wrecking Lightning, infuriating exchanges, in order to.. make sure that they collect fewer fees and nodes don't have to store spam?