We prove a special case of the Kakeya maximal function conjecture in $\mathbb{R}^3$ , with $C_{\epsilon}\delta^{-\epsilon}$ loss, when the centre lines of the tubes are $SL_2$ lines and the tubes satisfy a 2-dimensional ball condition (implied by the Wolff axioms). We show that the exponent $p = 3/2$ is sharp and that some loss (such as $C_{\epsilon}\delta^{-\epsilon}$) is necessary, even in the $SL_2$ case where the $\delta$-tubes have $\delta$-separated directions and the cardinality of the tube family is maximal ($∼ \delta^{-2}$). The $SL_2$ Kakeya maximal inequality is deduced from an $L^{3/2}$ inequality for restricted families of projections onto planes. A related $L^{3/2-\epsilon}$ inequality is also derived for restricted projections onto lines, and an application is given to generic intersections of sets in $\mathbb{R}^3$ with "light rays" and "light planes".