Context : In 2012 , we applied a three-dimensional formulation to kinematic measurements of the Galactic thick disk and derived a surprisingly low dark matter density at the solar position .
This result was challenged by Bovy & Tremaine ( 2012 , ApJ , 756 , 89 ) , who claimed that the observational data are consistent with the expected local dark matter density if a one-dimensional approach is adopted .

Aims : We aim at clarifying whether their work definitively explains our results , by analyzing the assumption at the bases of their formulation and their claim that this returns a lower limit for the local dark matter density , which is accurate within 20 % .

Methods : We find that the validity of their formulation depends on the underlying mass distribution .
We therefore analyze the predictions that their hypothesis casts on the radial gradient of the azimuthal velocity \partial _ { R } \overline { V } and compare it with observational data as a testbed for the validity of their formulation .

Results : We find that their hypothesis requires too steep a profile of \partial _ { R } \overline { V } ( Z ) , which is inconsistent with the observational data both in the Milky Way and in external galaxies .
As a consequence , their results are biased and largely overestimate the mass density .
Dynamical simulations also show that , contrary to their claims , low values of \partial _ { R } \overline { V } are compatible with a Milky Way-like potential with radially constant circular velocity .
We nevertheless confirm that , according to their criticism , our assumption \partial _ { R } \overline { V } =0 is only an approximation .
If this hypothesis is released , and the available information about \partial _ { R } \overline { V } in the thick disk is used , the resulting local dark matter density increases by a tiny amount , from 0 \pm 1 to 2 \pm 3 mM _ { \odot } pc ^ { -3 } , with an upper limit of \sim 3.5 mM _ { \odot } pc ^ { -3 } .
Hence , this approximation has negligible influence on our results .

Conclusions : Our analysis shows that their criticism is not a viable explanation for the inferred lack of dark matter at the solar position detected by us .
More studies are required to understand these unexpected results .