  We consider the dynamical model of a binary bosonic gas trapped in a
symmetric dual-core cigar-shaped potential. The setting is modeled by a system
of linearly-coupled one-dimensional Gross-Pitaevskii equations with the cubic
self-repulsive terms and quadratic attractive ones,which represent the
Lee-Huang-Yang corrections to the mean-field theory in this geometry. The main
subject is spontaneous symmetry breaking (SSB) of quantum droplets (QDs),
followed by restoration of the symmetry, with respect to the symmetric
parallel-coupled trapping cores, following the increase of the QD's total norm.
The SSB transition and inverse symmetry-restoring one form a bifurcation loop,
whose shape in concave at small values of the inter-core coupling constant,
$\kappa$, and convex at larger $\kappa $. The loop does not exist above a
critical value of $\kappa $. At very large values of the norm, QDs do not break
their symmetry, featuring a flat-top shape. Some results are obtained in an
analytical form, including an exact front solution connecting constant zero and
finite values of the wave function. Collisions between moving QDs are
considered too, demonstrating a trend to merger into breathers.
