Left-half-plane conjecture for third-order Stirling subset polynomials

For $n\geq 0$, let $s_{3,n}(x)$ denote the third-order Stirling subset polynomial. Third-order Stirling subset zero conjecture. The zeros of $s_{3,n}(x)$ lie in the left half-plane. This is suggested by plots in the paper and is stated as an open conjecture; the broader asymptotic zero-distribution problem is also left open.