Assume an odd perfect number exists could be it written as $x^3+y^3+z^3$?

Question

Touchard $(1953)$ proved that an odd perfect number, if it exists, must be of the form $12k+1$ or $ 36k+9$(Holdener $2002 $) , A necessary condition for any number to be written as a cubic sum $x^3+y^3+z^3$ is that number can't be $4$ or $5 \bmod 9$,one can think about $12k+1$ which is $4$ or $7$ mod $9$ then we can't say anything since we have the remainder is equal to $4$, but if it is of the form $ 36k+9$ this is conguent $1 \bmod 4$, Which is impossible ? , Now I ask Assume an odd perfect exists could be it written as $x^3+y^3+z^3$ ?