On the Tits alternative for generalized tetraedron groups of type $(2, 2, N, 2, 2, 2)$

Generalized tetraedron groups have a presentation of the form
$$\Gamma=\left\langle x_1,x_2,x_3\mid x_1^{k_1}=x_2^{k_2}=x_3^{k_3}=R_{12}(x_1,x_2)^l=R_{23}(x_2,x_3)^m=R_{13}(x_1,x_3)^n=1\right\rangle.$$
There exists a Rosenberger’s conjecture that the Tits alternative holds for generalized tetrahedron groups. This conjecture is open for groups of the form $\left\langle x_1,x_2,x_3\mid x_1^{k_1}=x_2^{k_2}=x_3^{k_3}=R_{12}(x_1,x_2)^2=(x_1^\alpha x_3^\beta)^2=(x_2^\gamma x_3^\delta)^2=1\right\rangle$, $\frac1{k_1}+\frac1{k_2}+\frac1{k_3}\geqslant\frac12$. In this paper, a number of sufficient conditions are found for fulfillment the Tits alternative for groups
$$\Gamma=\left\langle a,b,c\mid a^2=b^n=c^2=R(a,b)^2=(b^\alpha c)^2=(ac)^2=1\right\rangle.$$