This is a follow-up to this question by Marc Palm asked 7 years ago:

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $G$ a reductive group over $K$. Is every irreducible cuspidal representation induced from an open, compact-mod-center subgroup?

My specific questions are:

Question 1:Have there been any new results on this question in the last 7 years?

(Meta remark: I don't know of a different possibility to draw attention to an MO question again, asking for new results, other than asking the question again.)

Question 2:Do people generally believe that this is true, or are they expecting that there are some weird counterexamples somewhere?

This is of course a very hazy question, but sometimes there is a common "folklore" belief in a community about some open questions. I don't belong to the community of experts on this, that's why I'm asking.

Question 3:I once heard the name Bernstein Conjecture for the conjecture that the answer is "yes". Is this a common name?