Conjecture for Stirling cycle numbers of all orders

Let $C^{(r)}$ be the Stirling cycle triangle of order $r$, let

$$\widecheck{C}^{(r)}=\left(C^{(r)}\right)^{\rm rev}$$

be the triangle formed by reversing the rows of $C^{(r)}$, and let $c_{r,n}(x)$ be its row-generating polynomials. A sequence of polynomials is coefficientwise Hankel-totally positive in $x$ when every minor of its Hankel matrix has coefficients that are nonnegative in $x$. Conjecture for Stirling cycle numbers of all orders. The following are true: (a) $C^{(r)}$ is totally positive for all $r\geq 1$. (b) $\widecheck{C}^{(r)}$ is totally positive for $r=1,2$ and is not totally positive when $r\geq 3$. (c) The polynomials $c_{r,n}(x)$ are negative-real-rooted for $r=1,2$ and have non-real complex zeros for $r\geq 3$ and $n\geq 3$. Also, the row sequences of $C^{(r)}$ are log-concave for $1\leq r\leq 5$. (d) The sequence $(c_{r,n}(x))_{n\geq 0}$ is coefficientwise Hankel-totally positive in $x$ for all $r\geq 1$. These assertions refine the known low-order relationship between Stirling cycle triangles and Eulerian-type triangles; the cases $r\geq 3$ include both failures of row-reversed total positivity and non-real-rootedness, while the total positivity of $C^{(r)}$ and the Hankel property are asserted for every order.