What are the First Amendment obstacles to mandatory front-of-pack labeling?

Suppose the government wanted to require front-of-pack nutrition labels for packaged food and beverages, making it easier for consumers to see at a glance some key nutritional qualities.

In the United States, could the manufacturers claim that such a rule violates their First Amendment rights?

In the journal Food Policy this past summer, Jennifer Pomeranz, Dariush Mozaffarian, Renata Micha, and I study the precedents. Much depends on whether a particular labeling policy could satisfy a legal principle called the Zauderer test. This test stipulates, for example, that the mandatory labeling information must be factual and uncontroversial. 

Looking at the wide array of front-of-pack labeling schemes around the world, we find that some varieties are more likely than others to pass this test. Certain proposals for simple mandatory symbols indicating "healthy" or "unhealthy" broadly, without communicating much nutrition information, could be ruled unconstitutional.

The article is titled, "Mandating front-of-package food labels in the U.S. – What are the First Amendment obstacles?"

Pomeranz et al. (2019). Table 1 (small excerpt).