Conjecture on the zeroes of the Lah generating polynomial

Let $P_{n,k}(t)$ be the generating polynomial associated with the Lah distribution. Negative-real-part zeroes conjecture. All complex zeroes of $P_{n,k}$ have negative real parts. Numerical simulations show that the zeroes are generally not real, except when $k=1$, motivating this conjecture as a possible replacement for real-rootedness in applications of the Harper method. Its status beyond the reported numerical evidence is open.