In the context of bouncers (see the bouncers write-up: here) we define that a formula tape f' is a special case of formula tape f if if f' can be obtained from f by replacing subwords of the form (w) by w^n(w)w^m for some n,m\geq 0 and w\in\Sigma^+ (1).

However, a more general and abstract definition of special case for regular languages is: \mathcal{L}(f') \subseteq \mathcal{L}(f) (2).

Clearly, (1) \Rightarrow (2). We believe that (2) \Rightarrow (1) with the additional assumptions (that come for free with bouncers) that the formula tapes are aligned and contain the same number of repeaters and that |r_i| = |r'_i|, i.e. corresponding repeaters in each formula tape are of same size.

Here is a proof given by @savask for the case of 1 repeater:

Nonetheless, (2) \Rightarrow (1) remains open in the general case (arbitrary number of repeaters) and it would be cool to have a proof!