On prefix palindromic length of automatic words

The prefix palindromic length PPLu(n) of an infinite word u is the minimal number concatenated palindromes needed to express n u. Since 2013, it still unknown if unbounded for every aperiodic u, even though this has been proven almost all words. At same time, only well-known nontrivial which function precisely computed Thue-Morse t. This 2-automatic and, predictably, its PPLt(n) 2-regular, but case automatic words? In paper, we prove that k-regular k-automatic containing a finite palindromes. For two such words, namely paperfolding and Rudin-Shapiro word, derive formula function. Our computational experiments suggest generally not true: period-doubling does look Fibonacci Fibonacci-regular. If proven, these results would give rare (if first) examples natural regular.