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

Let $f_{0}(z)=z^{2}+1/4$ and ${\mathcal E}_{0}$ 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 $\mathbb R/ \mathbb Z$. We prove that if $\overline{\sigma}_{n}\in{\mathcal E}_{0}$converges to $\overline{\sigma}\in\partial{\mathcal E}_{0}$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 $J(f_{0},g_{\sigma})$.