Sign-alternation conjecture for the flat Bernoulli game coefficients

Let $a_n$ be the coefficient of $t^n/n!$ in $1/(1-\ln(1+t))$, so that for $n\geq 1$,

$$a_n=(-1)^n\left(K_n-nK_{n-1}\right)=(-1)^n\frac{\kappa_{n+1}-(n+1)\kappa_n}{n+1}.$$

The first $11$ entries are positive, while $a_{12}$ is negative and the subsequent entries appear to have alternating signs. Sign-alternation conjecture. For $n\geq 12$,

$$n\kappa_{n-1}>\kappa_n.$$

Equivalently, $nK_{n-1}>K_n$ holds for $n\geq 11$. This is the conjectured continuation of the observed alternation in the coefficient sequence associated with the flat Bernoulli game; the supplied text gives no resolution of the conjecture.