mg.metric geometry - Malfatti Circles - Limiting point

"Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles" (for a brief historical account on this topic, see here and here on MathWorld).

Consider the triangle formed by the centers of these circles, one can draw a new set of smaller Malfatti circles in this triangle. What is the limiting point of this process?

One thing sort of discouraging is that I tried on an isosceles triangle, unfortunately did not find the limiting point matching any of the known relevant points (e.g., incenter or the first Ajima-Malfatti point).