  Let $G$ be a complex reductive group and $\mathcal{X}_{r}G$ denote the
$G$-character variety of the free group of rank $r$. Using geometric methods,
we prove that $E(\mathcal{X}_{r}SL_{n})=E(\mathcal{X}_{r}PGL_{n})$, for any
$n,r\in\mathbb{N}$, where E(X) denotes the Serre (also known as E-) polynomial
of the complex quasi-projective variety $X$, settling a conjecture of
Lawton-Mu\~noz in [LM]. The proof involves the stratification by polystable
type introduced in [FNZ], and shows moreover that the equality of E-polynomials
holds for every stratum and, in particular, for the irreducible stratum of
$\mathcal{X}_{r}SL_{n}$ and $\mathcal{X}_{r}PGL_{n}$. We also present explicit
computations of these polynomials, and of the corresponding Euler
characteristics, based on our previous results and on formulas of
Mozgovoy-Reineke for $GL_{n}$-character varieties over finite fields.
