  Chromonic liquid crystals are lyotropic materials which are attracting
growing interest for their adapatbility to living systems. To describe their
elastic properties, the classical Oseen-Frank theory requires anomalously small
twist constants and (comparatively) large saddle-splay constants, so large as
to violate one of Ericksen's inequalities, which guarantee that the Oseen-Frank
stored-energy density is bounded below. While such a violation does not prevent
the existence and stability of equilibrium distortions in problems with fixed
geometric confinement, the study of free-boundary problems for droplets has
revealed a number of paradoxical consequences. Minimizing sequences driving the
total energy to negative infinity have been constructed by employing ever
growing needle-shaped tactoids incorporating a diverging twist [Phy. Rev. E
106, 044703 (2022)]. To overcome these difficulties, we propose here a novel
elastic theory that extends for chromonics the classical Oseen-Frank stored
energy by adding a quartic twist term. We show that the total energy of
droplets is bounded below in the quartic twist theory, so that the known
paradoxes are ruled out. The quartic term introduces a phenomenological length
$a$ in the theory; this affects the equilibrium of chromonics confined within
capillary tubes. Use of published experimental data allows us to estimate $a$.
