Question about Type IIB supergravity equations of motion

For a target space $p$-form in local coordinates

$$F_p~=~\frac{1}{p!}F_{\mu_1\mu_2\ldots\mu_p}~\mathrm{d}x^{\mu_1}\wedge \mathrm{d}x^{\mu_2}\wedge\ldots\wedge \mathrm{d}x^{\mu_p},\tag{1}$$

the paper defines a scalar

$$F_p^2~:=~F_{\mu_1\mu_2\ldots\mu_p}~g^{\mu_1\nu_1}~g^{\mu_2\nu_2}\ldots g^{\mu_p\nu_p}~F_{\nu_1\nu_2\ldots\nu_p},\tag{2}$$

and a symmetric covariant tensor

$$(F_p^2)_{\mu_1\nu_1}~:=~F_{\mu_1\mu_2\ldots\mu_p}~g^{\mu_2\nu_2}\ldots g^{\mu_p\nu_p}~F_{\nu_1\nu_2\ldots\nu_p}.\tag{3}$$

$F_{n-1}$'s are curvatures of higher gauge fields $A_{n-1}$'s, so they are n-forms. The expression $(F_{n}^2)_{\mu \nu}$ means schematically $F_{\mu \rho_1 \cdots \rho_{n-1}} F_{\nu}^{\phantom{a} \rho_1 \cdots \rho_{n-1}}$, where the indices are risen with the inverse metric $g^{\mu \nu}$.