Explicit Increasing Degree function with inverse Ackermann complexity

Construct an explicit function f_{ack}:\mathbb{N}\to\mathbb{N} such that the associated Increasing Degree problem \Pi_{f_{ack}} has round complexity equal to the inverse Ackermann function \alpha(n) in the LOCAL model.

Background

The paper realizes several levels of the inverse-Ackermann hierarchy through Increasing Degree problems: functions growing linearly, exponentially, and as iterated exponential towers yield complexities corresponding to \alpha_2(n), \alpha_3(n), and \alpha_4(n), respectively.

It remains unresolved how to define a single explicit function whose iterates produce the full inverse Ackermann complexity \alpha(n), rather than a fixed level \alpha_k(n). The requested construction is presented as a concrete instance of the broader problem of systematically realizing computable complexity functions.