I don't see Louisiana building I-69 north of I-20 in my lifetime. But I can see I-69 being built from the Texas border up to I-20 in the next 20-25 years.
I actually see I-69 / 369 built all of the way to Texarkana in that time frame. On the other hand, the part from I-20 (Haughton) to Memphis I really don't see happening any time soon. Maybe in the next fifty years. Maybe not.
I am not sure I see the part from the I-69 / 369 split to I-49 built that soon either. The part from I-49 to I-20 might get built a little more quickly.
Once the section from I-49 to I-20 is completed (hopefully by the 2040s), Louisiana will have a reason to work with Texas to fill in the gap between the I-69/I-369 split and I-49.
By then I-369 should be nearing completion along with the three I-69 southern branches. But Texas rightfully shouldn't make any move on its last 15 or so miles until both states have decided where the border crossing will be located.