Continuity of Hausdorff dimension of Julia-Lavaurs sets as a function of the phase

Let $ f_{0}(z)=z^{2}+1/4$ and $\Sigma $ the set of phases $\overline{\sigma \ }$ such that the critical point $ 0$ escapes in one step by the Lavaurs map $ g_{\sigma }$; it is a topological strip in the cylinder of phases whose boundary consists of two Jordan curves symmetric wrt $ \RR /\ZZ $. We prove that if $ \overline{\sigma }_{n}\in \Sigma \ $converges to $ \overline{\sigma }\in \partial \Sigma $ in such a way that $ g_{\sigma _{n}}(0)$ converges to $ g_{\sigma }(0)$ along an external ray then the Hausdorff dimension of the Julia-lavaurs set $ J(f_{0},g_{\sigma_{n}})$ converges to the Hausdorff dimension of $J(f_{0},g_{\sigma })$.