Real-rootedness, log-concavity, and Frankl–Füredi–Kalai conjecture for zig-zag Eulerian polynomials

For $n\geq 1$, write $Z_n(t)=\sum_k z(n,k)t^k$, and let

$$\gamma_n=(\gamma_{n,0},\gamma_{n,1},\ldots,\gamma_{n,\lfloor (n-2)/2\rfloor})$$

be the gamma-vector associated with $Z_n(t)$. A sequence is log-concave when $a_k^2\geq a_{k-1}a_{k+1}$ wherever the terms are defined, and a vector satisfies the Frankl–Füredi–Kalai inequalities when it is the $f$-vector of a balanced simplicial complex. Zig-zag polynomial conjecture. For all $n\geq 1$: (a) $Z_n(t)$ has only real roots; (b) the sequence $\{z(n,k)\}_{0\leq k\leq n-2}$ is log-concave, namely

$$z(n,k)^2\geq z(n,k-1)z(n,k+1)$$

for each $k\in\{1,2,\ldots,n-3\}$; and (c) $\gamma_n$ satisfies the Frankl–Füredi–Kalai inequalities, equivalently, $\gamma_n$ is the $f$-vector of a balanced simplicial complex. These are posed as open questions for the zig-zag Eulerian polynomials, alongside the general relationship between gamma-nonnegativity, log-concavity, and real-rootedness.