Complexity and palindromic defect of infinite words

In this note, we state a conjecture, and prove it in the periodic case, which is an equality relating the number of factors and palindromic factors of infinite words. This equality establishes a link between two inequalities, one due to Droubay, Justin and Pirillo, and the other to Baláži, Masáková and Pelantová, by means of the palindromic defect.