Existence of arbitrarily deep preperiod reductions if the lower-bound conjecture fails

Determine whether the following implication holds: if there exist an odd prime p and an integer r≥2 with s(p^r)<r−2, then for every integer k≥3 there exist an odd prime p and an integer r≥k such that s(p^r)=r−k.

Background

The paper presents a second conjecture concerning the structure of possible failures of the lower bound s(p^r)≥r−2. The preceding computational evidence found examples with reductions of one and two, namely s(p^r)=r−1 and s(43^980)=r−2, but no example with reduction three or more in the tested range.

The proposed implication asserts that any failure below r−2 would not be isolated: it would entail examples with arbitrarily large reductions r−s(p^r), occurring at exponents r at least k for every k≥3. The authors explicitly leave this conjecture unresolved.