Conjecture on the imaginary zeros and separation of Stirling permutation polynomials

Let $T_n(x)$ be the polynomial defined by the exponential generating function

$$T(x,z)=\sum_{n=0}^\infty T_n(x)\frac{z^n}{n!}.$$

For polynomials with only real coefficients and only imaginary zeros, say that $f(x)$ separates $F(x)$ if $\deg F=\deg f+2$ and the ordered real and imaginary parts of the zeros of $f(x)$ separately interlace those of $F(x)$. Imaginary-zero and separation conjecture. For $n\geq 2$, all zeros of $T_n(x)/x$ are imaginary and $T_n(x)/x$ separates $T_{n+1}(x)/x$. This conjecture predicts a real-root-analogue property for the polynomials $T_n(x)$, extending the known real-zero and separation properties of the related polynomials $R_n(x)$.