Spherical functions of symmetric forms and a conjecture of Hironaka

For all $r\ge 1,$ we verify the following conjecture of Hironaka: for a $p$-adic field $F$ with $p$ odd, the space of spherical functions of ${\mathrm{S}\mathrm{y}\mathrm{m}}_{r\times r}(F)\cap {\mathrm{G}\mathrm{L}}_r(F)$ is free of rank $4^r$ over the Hecke algebra.