  For all $r\ge1,$ we verify the following conjecture of Hironaka: for a
$p$-adic field $F$ with $p$ odd, the space of spherical functions of
$\mathrm{Sym}_{r\times r}(F)\cap\mathrm{GL}_r(F)$ is free of rank $4^r$ over
the Hecke algebra.
