This is unlikely.

If there are exactly 1 million participants in the LN network, in order for every participant to be able to make a payment to any other participant in the network using no more than 6 "hops" (the last hop being the person you are paying), then each participant must, on average have at least 6 open channels with distinct participants.

If Alice needs to pay Bob, and the shortest route between the two is 6 hops, then all 6 LN nodes must be online at the same time, and if any one of the LN nodes are not online (or otherwise fails to sign the necessary transactions), the payment will fail. If LN resembles anything close to a mesh network, the majority of the time participants will be unable to transact with the majority of the network if you assume each LN node has 67% uptime (each node is turned off for 8 hours per day -- a mesh network would mean that most LN nodes would be compromised of "home users").

If the LN network were to resemble something closer to a 'hub and spoke' network, the issue of LN nodes downtime would not be as prevalent, however there would be other issues. Each hub would have an outsized influence over the network, and would have the ability to wreck havoc on the network either by raising fees substantially, censoring transactions of those directly connected to each hub, and downtime of a small number of hubs would have the potential to fragment the entire LN network. Also, running a hub node would be incredibility risky because private keys would need to be kept effectively 100% online (on a computer connected to many other people), and the hub's private keys must control a large amount of bitcoin. If you assume that hubs collectively control 25% of bitcoin, then hubs must collectively keep ~$34 billion worth of bitcoin online.