The paper "A study of certain modular representations" (ac.els-cdn.com/0021869378901163/…) proves a lot of results concerning the structure of $\text{Sym}^{g}(k^2)$, among other things that $\text{Sym}^{g+p (p-1)}(k^2) \cong \text{Sym}^g(k^2) \oplus \text{projective module}$.

Dear Mikhail! I added a sketch of how to do the computation by hand and the Magma code I used. Let me know if I need to add more detail.

I used Magma, but the example is small enough that one can do it by hand. A reference for the relevant functions in Magma is magma.maths.usyd.edu.au/magma/handbook/… It is also possible to use GAP, see gap-system.org/Manuals/doc/ref/chap39.html#X7CA0B6A27E0BE6B8

For $n\geq \lvert b\rvert$, the number $a\times n!+b$ is divisible by $b$, so if $\lvert b\rvert\geq 2$ the answer is yes :-)

@StanleyYaoXiao I dont think this is correct. Zhangs's thesis can be found here. As far as I can see he doesnt claim to prove the Jacobian conjecture.