I think there may be a useful distinction here between prior restraint and compelled speech. If you post the statement "we have not received a gagged warrant since January 3 2023" and you receive a gagged warrant on January 4 2023, does the government really have the power to compel you to post the statement "we have not received a gagged warrant since January 5 2023" the next day? You don't even have to take down the January 3 canary; just leave it up long enough that people get suspicious that you haven't updated it.

The government routinely compels speech.

Can you give an example where the US government forced a company to lie to the public which we've only found out later?

Not just a lie, but false advertising, which is a federal offense in and of itself.

> The problem with this analysis is that First Amendment rights were already successfully infringed with the gag order that necessitated the canary, which itself has to meet the (maximal) strict scrutiny standard

Has the supreme court actually ruled on the constitutionality of gag orders? Until then it's not completely settled that it's despite rulings in lower courts.

These specifically, not that I know of. But gag orders in general? Nebraska Press v. Stuart, 3 part test: harm to the person being gagged, least restrictive means, and effectiveness of the order. That was in a press case, where the burden is much higher on the government than in these cases.

Specific gag orders can and have failed in appeals!

The problem with this argument as it pertains to warrant canaries is that defeating the gag order also defeats the purpose of the warrant canary. The question we're begging here is: if the gag order survives strict scrutiny, why won't the order to keep the canary up?