  In a recent experiment on the interlayer magnetoresistance in the
quasi-two-dimensional organic salt, $\alpha$-(BEDT-TTF)$_2$I$_3$, it has been
observed that at low temperatures, interlayer tunneling attains phase
coherence, leading to the emergence of a three-dimensional electronic
structure. Theoretically and experimentally it has been suggested that the
system exhibits characteristics of a three-dimensional Dirac semimetal as a
consequence of broken time-reversal symmetry and inversion symmetry. Here, we
perform a theoretical calculation of the magnetoconductivity under an in-plane
magnetic field and demonstrate that the system displays a planar Hall effect.
Our calculations are based on a realistic model for
$\alpha$-(BEDT-TTF)$_2$I$_3$ incorporating interlayer tunneling and the tilt of
the Dirac cone. Given that the planar Hall effect is anticipated as a
consequence of chiral anomaly, our findings provide support for the
classification of $\alpha$-(BEDT-TTF)$_2$I$_3$ as a three-dimensional Dirac
semimetal.
