Almost sure convergence at the boundary of the complex phase

Prove that, as the complex parameter z approaches 1 within the closed region P_1\cup P_{1,2}, the normalized complex additive-martingale limit W_\infty(z)/(1-z) converges almost surely to 2Z_\infty.

Background

The paper proves almost sure convergence of W_\infty(z)/(1-z) to 2Z_\infty when z approaches 1 non-tangentially inside P_1, and proves convergence in probability when z approaches 1 through P_1\cup P_{1,2}. The authors’ method does not establish almost sure convergence for tangential approaches or along the boundary P_{1,2}; the conjecture asks for this stronger boundary result.