Real-rootedness of the Eulerian transformation on the binomial basis cone

Let $n\geq 0$, and let $f$ be a polynomial of the form

$$f=\sum_{k=0}^n a_k t^k(1+t)^{n-k},\qquad a_k\geq 0\quad(0\leq k\leq n).$$

Let $\operatorname*{\mathcal E}:\mathbb{R}[t]\to\mathbb{R}[t]$ be the Eulerian transformation defined by $\operatorname*{\mathcal E}(t^m)=A_m(t)$, where $A_m(t)$ is the $m$-th Eulerian polynomial. The paper's conjecture. The polynomial $\operatorname*{\mathcal E}(f)$ is real-rooted.

This conjecture seeks a restricted real-rootedness-preservation property after Brenti's broader conjecture was disproved. The paper provides partial evidence, including alternatingly increasing and hence unimodal coefficients for the transformed polynomials, as well as real-rootedness results for important subfamilies; the full assertion remains open in the supplied text.