Stability conjecture for the edge points of linear steps

Let $a,b\geqslant 1$ be real numbers, and for $N\geqslant 3$ let $c_b^N(a)$ denote the stabilized symplectic embedding capacity, while $c_b(a)$ denotes the corresponding four-dimensional capacity. The edge points are the values of $a$ occurring at the ends of the linear steps in the graph of $c_b(a)$. Stability conjecture. At every edge point of a linear step, one has

$$c_b^N(a)=c_b(a)\quad\text{for all }N\geqslant 3.$$

The preceding folding argument gives an upper bound for the stabilized capacity, and the conjecture asserts that stabilization does not improve the capacity at these distinguished points. The supplied text gives no resolution status.