  Kaplansky conjectured that if two positive-definite real ternary quadratic
forms have perfectly identical representations over $\mathbb{Z}$, they are
constant multiples of regular forms, or is included in either of two families
parametrized by $\mathbb{R}^2$ (so called, hexagonal and rhombohedral
families). Our results aim to clarify the limitations imposed to such a pair by
computational and theoretical approaches. Firstly, the result of an exhaustive
search for such pairs of integral quadratic forms is presented, in order to
provide a concrete version of the Kaplansky conjecture. The obtained list
contains a small number of non-regular forms that are confirmed to have the
identical representations up to 3,000,000, although a strong limitation on the
existence of such pairs is still observed, regardless of whether the
coefficient field is $\mathbb{Q}$ or $\mathbb{R}$. Secondly, we prove that if
two pairs of ternary quadratic forms have the identical simultaneous
representations over $\mathbb{Q}$, their constant multiples are equivalent over
$\mathbb{Q}$. This was motivated by the question why the other families were
not detected in the search. In the proof, the parametrization of quartic rings
and their resolvent rings by Bhargava is used to discuss pairs of ternary
quadratic forms.
