  We study quantum phase transitions in the asymmetric variation of the
three-leg Heisenberg tube for half-odd-integer spin, with a modulation of one
of the rung exchange couplings $J'_\perp$ while the other two are kept constant
$J_\perp$. We focus on the strong rung-coupling regime $J_\perp \gg
J_\parallel$, where $J_\parallel$ is the leg coupling, and analyze the
effective spin-orbital model with a transverse crystal field in detail.
Applying the Abelian bosonization to the effective model, we find that the
system is in the dimer phase for the general half-odd-integer-spin cases
without the rung modulation; the phase transition between the dimer and
Tomonaga-Luttinger-liquid phases induced by the rung modulation is of the
SU(2)-symmetric Berezinskii-Kosterlitz-Thouless type. Moreover, we perform a
level spectroscopy analysis for the effective model for spin-1/2 using exact
diagonalization, to determine the precise transition point $| J'_\perp -
J_\perp| /J_\parallel \sim 0.283$ in the strong rung-coupling limit. The
presence of the dimer phase in a small but finite region is also confirmed by a
density-matrix renormalization group calculation on the original spin-tube
model.
