Strictly nef divisors on singular threefolds

Let $X$ be a normal projective threefold with mild singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. First, we show the ampleness of $K_X+tL_X$ with sufficiently large $t$ if either the Kodaira dimension $\kappa(X)\neq 0$ or the augmented irregularity $q^{\circ}(X)\neq 0$. Second, we show that, if $(X,\Delta)$ is a projective klt threefold pair with the anti-log canonical divisor $-(K_X+\Delta)$ being strictly nef, then $X$ is rationally connected.