On the equation $F(n^3)=F(n^3-1)+D$ and some conjectures

We prove that if the complex number $D$ and the completely multiplicative function $F$ satisfy the equation $F(n^3)=F(n^3-1)+D$ for every positive integer $n>1$, then $F$ is the identity function if $D\neq 0$. In the case $D=0$, there are two solutions $F$. We also state three conjectures and prove some partial results.