  The generic Hidden Local Symmetry (HLS) model has given rise to the BHLS$_2$
variant which introduces symmetry breaking in the vector meson sector; the
central mechanism is a modification of the covariant derivative at the root of
the HLS approach. However, the description of the Belle $\tau$ dipion spectra
is not satisfactory whereas the dealing with its annihilation sector ($e^+ e^-
\rightarrow \pi^+ \pi^-/\pi^+ \pi^-\pi^0/ \pi^0 \gamma/\eta \gamma/K^+ K^-/K_L
K_S$) is optimum. This issue is solved by an additional breaking term allowing
to include the $[\pi^0,\eta,\eta^\prime]$ system. This extension of the usual
't~Hooft determinant term only affects the BHLS$_2$ kinetic energy term. One
obtains a fair account for the $\tau$ dipion spectra as well as for the
annihilation channels.
  The Belle dipion spectrum provides evidence for a violation of CVC in the
$\tau$ lepton decay; this evidence is strengthened by imposing the conditions
$<0|J_\mu^q |[q^\prime \overline{q^\prime}](p)>=ip_\mu f_q \delta_{q q^\prime
},
  \{ [q \overline{q}], q=u,d,s\}$ on BHLS$_2$ axial current matrix elements.
BHLS$_2$ is found to recover the usual (completed) formulae for the
[$\pi^0,~\eta,~\eta^\prime$] mixing parameters and the global fits return
mixing parameter values in agreement with expectations and better
uncertainties. Updating the muon HVP, one argues that the strong tension
between the KLOE and BaBar pion form factors imposes to provide two solutions :
$a_\mu^{HVP-LO}({\rm KLOE})=687.31 \pm 2.93$ and $a_\mu^{HVP-LO}({\rm
BaBar})=692.36 \pm 2.95$, in units of $10^{-10}$, rather than some combination
of these. Taking into account common systematics, their differences with the
experimental BNL-FNAL average value exhibit significances $> 5.4\sigma$ (KLOE)
and $> 4.1\sigma$ (BaBar) with fit probabilities favoring the former.
