By the rules-as-written, is a player character required to engage in downtime activity?
Another way to ask the question is (per @SorcererQzot's comment) "Do characters actually have to stop adventuring between adventures?" since this question is based the discussions at the link.
Downtime Activities (p. 70, Basic Rules)
Between adventures, the DM might ask you what your character is doing during his or her downtime. Periods of downtime can vary in duration, but each downtime activity requires a certain number of days to complete before you gain any benefit, and at least 8 hours of each day must be spent on the downtime activity for the day to count. The days do not need to be consecutive. If you have more than the minimum amount of days to spend, you can keep doing the same thing for a longer period of time, or switch to a new downtime activity. (The listed pursuits are crafting, recuperating, practicing a profession, researching, training).
There are a variety of things that a character can do during downtime, but is the actual taking of downtime required?
Taking a Short Rest is a period of downtime that seems to be an exception to the general case made above: if you want a Short Rest you have to be in a special "downtime" (at least 1 hour long per Basic Rules, p. 70) but it can occur in the middle of an adventure, and in the middle of the adventuring day.
Beyond that "in the field" downtime (which looks like an outlier) what is the hard requirement for downtime?
Answers must cite the rules as written, since the original question was raised in a pure RAW context.
Why is the Short Rest an outlier? Because "Adventurers can take short rests in the midst of an adventuring day" while the bulk of Downtime Activity happens between adventuring days. (p. 70 - 71, Basic Rules).